# Find the Local Maxima and Minima y=e^(2x)-e^x Rewrite the equation as a function of .
Find the first derivative of the function.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Exponential Rule which states that is where =.
Find the second derivative of the function.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Exponential Rule which states that is where =.
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor the left side of the equation.
Rewrite as .
Let . Substitute for all occurrences of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Replace all occurrences of with .
Replace the left side with the factored expression.
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
The equation cannot be solved because is undefined.
Undefined
There is no solution for
No solution
No solution
Set the next factor equal to and solve.
Set the next factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Expand the left side.
Expand by moving outside the logarithm.
The natural logarithm of is .
Multiply by .
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.
Evaluate the second derivative.
Simplify each term.
Simplify by moving inside the logarithm.
Exponentiation and log are inverse functions.
Apply the product rule to .
One to any power is one.
Raise to the power of .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Exponentiation and log are inverse functions.
Simplify the expression.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
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
Find the y-value when .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Simplify by moving inside the logarithm.
Exponentiation and log are inverse functions.
Apply the product rule to .
One to any power is one.
Raise to the power of .
Exponentiation and log are inverse functions.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Subtract from .
Move the negative in front of the fraction.
The final answer is .
These are the local extrema for .
is a local minima
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