Find the Local Maxima and Minima y=e^(2x)-e^x (2024)

Popular Problems

Calculus

Find the Local Maxima and Minima y=e^(2x)-e^x

Step 1

Write as a function.

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of with respect to is .

Step 2.2

Evaluate .

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Step 2.2.1

Differentiate using the chain rule, which states that is where and .

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Step 2.2.1.1

To apply the Chain Rule, set as .

Step 2.2.1.2

Differentiate using the Exponential Rule which states that is where =.

Step 2.2.1.3

Replace all occurrences of with .

Step 2.2.2

Since is constant with respect to , the derivative of with respect to is .

Step 2.2.3

Differentiate using the Power Rule which states that is where .

Step 2.2.4

Multiply by .

Step 2.2.5

Move to the left of .

Step 2.3

Evaluate .

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Step 2.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.3.2

Differentiate using the Exponential Rule which states that is where =.

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of with respect to is .

Step 3.2

Evaluate .

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Step 3.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 3.2.2

Differentiate using the chain rule, which states that is where and .

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Step 3.2.2.1

To apply the Chain Rule, set as .

Step 3.2.2.2

Differentiate using the Exponential Rule which states that is where =.

Step 3.2.2.3

Replace all occurrences of with .

Step 3.2.3

Since is constant with respect to , the derivative of with respect to is .

Step 3.2.4

Differentiate using the Power Rule which states that is where .

Step 3.2.5

Multiply by .

Step 3.2.6

Move to the left of .

Step 3.2.7

Multiply by .

Step 3.3

Evaluate .

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Step 3.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 3.3.2

Differentiate using the Exponential Rule which states that is where =.

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to and solve.

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of with respect to is .

Step 5.1.2

Evaluate .

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Step 5.1.2.1

Differentiate using the chain rule, which states that is where and .

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Step 5.1.2.1.1

To apply the Chain Rule, set as .

Step 5.1.2.1.2

Differentiate using the Exponential Rule which states that is where =.

Step 5.1.2.1.3

Replace all occurrences of with .

Step 5.1.2.2

Since is constant with respect to , the derivative of with respect to is .

Step 5.1.2.3

Differentiate using the Power Rule which states that is where .

Step 5.1.2.4

Multiply by .

Step 5.1.2.5

Move to the left of .

Step 5.1.3

Evaluate .

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Step 5.1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 5.1.3.2

Differentiate using the Exponential Rule which states that is where =.

Step 5.2

The first derivative of with respect to is .

Step 6

Set the first derivative equal to then solve the equation .

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Step 6.1

Set the first derivative equal to .

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Rewrite as .

Step 6.2.2

Let . Substitute for all occurrences of .

Step 6.2.3

Factor out of .

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Step 6.2.3.1

Factor out of .

Step 6.2.3.2

Factor out of .

Step 6.2.3.3

Factor out of .

Step 6.2.4

Replace all occurrences of with .

Step 6.3

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Step 6.4

Set equal to and solve for .

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Step 6.4.1

Set equal to .

Step 6.4.2

Solve for .

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Step 6.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

Step 6.4.2.2

The equation cannot be solved because is undefined.

Undefined

Step 6.4.2.3

There is no solution for

No solution

No solution

No solution

Step 6.5

Set equal to and solve for .

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Step 6.5.1

Set equal to .

Step 6.5.2

Solve for .

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Step 6.5.2.1

Add to both sides of the equation.

Step 6.5.2.2

Divide each term in by and simplify.

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Step 6.5.2.2.1

Divide each term in by .

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Cancel the common factor of .

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Step 6.5.2.2.2.1.1

Cancel the common factor.

Step 6.5.2.2.2.1.2

Divide by .

Step 6.5.2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

Step 6.5.2.4

Expand the left side.

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Step 6.5.2.4.1

Expand by moving outside the logarithm.

Step 6.5.2.4.2

The natural logarithm of is .

Step 6.5.2.4.3

Multiply by .

Step 6.6

The final solution is all the values that make true.

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

Step 9

Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Simplify by moving inside the logarithm.

Step 10.1.2

Exponentiation and log are inverse functions.

Step 10.1.3

Apply the product rule to .

Step 10.1.4

One to any power is one.

Step 10.1.5

Raise to the power of .

Step 10.1.6

Cancel the common factor of .

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Step 10.1.6.1

Cancel the common factor.

Step 10.1.6.2

Rewrite the expression.

Step 10.1.7

Exponentiation and log are inverse functions.

Step 10.2

Simplify the expression.

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Step 10.2.1

Write as a fraction with a common denominator.

Step 10.2.2

Combine the numerators over the common denominator.

Step 10.2.3

Subtract from .

Step 11

is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

is a local minimum

Step 12

Find the y-value when .

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Step 12.1

Replace the variable with in the expression.

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Simplify by moving inside the logarithm.

Step 12.2.1.2

Exponentiation and log are inverse functions.

Step 12.2.1.3

Apply the product rule to .

Step 12.2.1.4

One to any power is one.

Step 12.2.1.5

Raise to the power of .

Step 12.2.1.6

Exponentiation and log are inverse functions.

Step 12.2.2

To write as a fraction with a common denominator, multiply by .

Step 12.2.3

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

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Step 12.2.3.1

Multiply by .

Step 12.2.3.2

Multiply by .

Step 12.2.4

Combine the numerators over the common denominator.

Step 12.2.5

Subtract from .

Step 12.2.6

Move the negative in front of the fraction.

Step 12.2.7

The final answer is .

Step 13

These are the local extrema for .

is a local minima

Step 14

Find the Local Maxima and Minima y=e^(2x)-e^x (2024)

FAQs

How to find local minima and local maxima? ›

To find local maximum or minimum, first, the first derivative of the function needs to be found. Values of x which makes the first derivative equal to 0 are critical points. If the second derivative at x=c is positive, then f(c) is a minimum. When the second derivative is negative at x=c, then f(c) is maximum.

What are the minimum and maximum values of y, e, x? ›

Why is there no maximum and minimum of f(x) =e^x? At first glance it looks like the 'minimum' is 0 , however, ex is never actually equal to 0 . The minimum of a function can't be 0 , if the function itself isn't ever 0 . So this function has no minimum value.

What is the local minima of a graph? ›

The local minimum value of a graph is the point where the graph changes from a decreasing function to an increasing function. The local minimum can also be defined in a specific interval.

What is the formula for local maxima? ›

A point (x = a) of a function f (a) is called a Local maximum if the value of f(a) is greater than or equal to all the values of f(x). Mathematically, f (a) ≥ f (a -h) and f (a) ≥ f (a + h) where h > 0, then a is called the Local maximum point.

How to solve maxima and minima problems? ›

Steps in Solving Maxima and Minima Problems
  1. Take the first derivative of the given function.
  2. Set f ′ ( x ) = 0 and solve for to find all critical points.
  3. Take the second derivative of the given function.
  4. Plug in the critical points from step into the second derivative.

How to find maximum and minimum? ›

We will set the first derivative of the function to zero and solve for x to get the critical point. If we take the second derivative or f''(x), then we can find out whether this point will be a maximum or minimum. If the second derivative is positive, it will be a minimum value.

How to find the minimum of a function? ›

You can find this minimum value by graphing the function or by using one of the two equations. If you have the equation in the form of y = ax^2 + bx + c, then you can find the minimum value using the equation min = c - b^2/4a.

What is the value of E in limits? ›

It can be proved mathematically that (1+1n)n does go to a limit, and this limiting value is called e. The value of e is 2.7182818283… .

How do you solve local minima? ›

The local minimum is found by differentiating the function and finding the turning points at which the slope is zero. The local minimum is a point in the domain, which has the minimum value of the function.

How to check for local minima? ›

Step 1: Find all of the intervals on the graph where the function is increasing and decreasing. Step 2: Find all of the points where the function changes from decreasing to increasing. Write these points as coordinate pairs. They are the local minima of the graph, and they will look like valleys in the graph.

How to find local and absolute maxima and minima? ›

Step 1: Identify any local maxima/minima, as well as the endpoints of the graph. Step 2: Determine the coordinates of all of these points. Whichever has the highest y-value is our absolute maximum, and whichever has the lowest y-value is our absolute minimum.

What is the formula for the local minima? ›

Here x = k, is a point of local minimum, if f'(k) = 0, and f''(k) > 0. The point at x= k is the local minimum, and f(k) is called the local minimum value of the function f(x).

How do you find local and absolute maxima and minima? ›

Step 1: Identify any local maxima/minima, as well as the endpoints of the graph. Step 2: Determine the coordinates of all of these points. Whichever has the highest y-value is our absolute maximum, and whichever has the lowest y-value is our absolute minimum.

What is the formula for maxima and minima? ›

Ans. Differentiation is used to discover the local maxima/minima for a one-variable function, f(x). When f (x) = 0, maxima and minima occur. If f (a) = 0 and f (a) < 0, x = an is a maximum; if f (a) = 0 and f (a) > 0, x = a is a minimum.

What is the rule for local maxima and minima? ›

Local maxima: If f'(x) changes sign from positive to negative as x increases via point c, then f(c) gives the maximum value of the function in that range. Local minima: If f'(x) changes sign from negative to positive as x increases via point c, then f(c) gives the minimum value of the function in that range.

References

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