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