Calculus Examples

Find the Local Maxima and Minima f(x)=e^(1-2x^2)
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3
Replace all occurrences of with .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.3
Add and .
Step 1.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.5
Differentiate using the Power Rule which states that is where .
Step 1.2.6
Multiply by .
Step 1.3
Simplify.
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Step 1.3.1
Reorder the factors of .
Step 1.3.2
Reorder factors in .
Step 2
Find the second derivative of the function.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Product Rule which states that is where and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Differentiate.
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Step 2.4.1
By the Sum Rule, the derivative of with respect to is .
Step 2.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.3
Add and .
Step 2.4.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.5
Differentiate using the Power Rule which states that is where .
Step 2.4.6
Multiply by .
Step 2.5
Raise to the power of .
Step 2.6
Raise to the power of .
Step 2.7
Use the power rule to combine exponents.
Step 2.8
Simplify the expression.
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Step 2.8.1
Add and .
Step 2.8.2
Move to the left of .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Multiply by .
Step 2.11
Simplify.
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Step 2.11.1
Apply the distributive property.
Step 2.11.2
Multiply by .
Step 2.11.3
Reorder terms.
Step 2.11.4
Reorder factors in .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
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Step 4.1
Find the first derivative.
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Step 4.1.1
Differentiate using the chain rule, which states that is where and .
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Step 4.1.1.1
To apply the Chain Rule, set as .
Step 4.1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.1.1.3
Replace all occurrences of with .
Step 4.1.2
Differentiate.
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Step 4.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.3
Add and .
Step 4.1.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.5
Differentiate using the Power Rule which states that is where .
Step 4.1.2.6
Multiply by .
Step 4.1.3
Simplify.
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Step 4.1.3.1
Reorder the factors of .
Step 4.1.3.2
Reorder factors in .
Step 4.2
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
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Step 5.1
Set the first derivative equal to .
Step 5.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.3
Set equal to .
Step 5.4
Set equal to and solve for .
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Step 5.4.1
Set equal to .
Step 5.4.2
Solve for .
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Step 5.4.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 5.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 5.4.2.3
There is no solution for
No solution
No solution
No solution
Step 5.5
The final solution is all the values that make true.
Step 6
Find the values where the derivative is undefined.
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Step 6.1
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 7
Critical points to evaluate.
Step 8
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 9
Evaluate the second derivative.
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Step 9.1
Simplify each term.
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Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.1.3
Simplify each term.
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Step 9.1.3.1
Raising to any positive power yields .
Step 9.1.3.2
Multiply by .
Step 9.1.4
Add and .
Step 9.1.5
Simplify.
Step 9.1.6
Multiply by .
Step 9.1.7
Simplify each term.
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Step 9.1.7.1
Raising to any positive power yields .
Step 9.1.7.2
Multiply by .
Step 9.1.8
Add and .
Step 9.1.9
Simplify.
Step 9.2
Subtract from .
Step 10
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 11
Find the y-value when .
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Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
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Step 11.2.1
Simplify each term.
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Step 11.2.1.1
Raising to any positive power yields .
Step 11.2.1.2
Multiply by .
Step 11.2.2
Add and .
Step 11.2.3
The final answer is .
Step 12
These are the local extrema for .
is a local maxima
Step 13