Calculus Examples

Find the Critical Points f(x)=e^(1-20x+5x^2)
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1.1
To apply the Chain Rule, set as .
Step 1.1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.1.3
Replace all occurrences of with .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.3
Add and .
Step 1.1.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.5
Differentiate using the Power Rule which states that is where .
Step 1.1.2.6
Multiply by .
Step 1.1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.8
Differentiate using the Power Rule which states that is where .
Step 1.1.2.9
Multiply by .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to and solve for .
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Step 2.3.1
Set equal to .
Step 2.3.2
Solve for .
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Step 2.3.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.3.2.2
The equation cannot be solved because is undefined.
Undefined
Step 2.3.2.3
There is no solution for
No solution
No solution
No solution
Step 2.4
Set equal to and solve for .
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Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
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Step 2.4.2.1
Add to both sides of the equation.
Step 2.4.2.2
Divide each term in by and simplify.
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Step 2.4.2.2.1
Divide each term in by .
Step 2.4.2.2.2
Simplify the left side.
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Step 2.4.2.2.2.1
Cancel the common factor of .
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Step 2.4.2.2.2.1.1
Cancel the common factor.
Step 2.4.2.2.2.1.2
Divide by .
Step 2.4.2.2.3
Simplify the right side.
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Step 2.4.2.2.3.1
Divide by .
Step 2.5
The final solution is all the values that make true.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
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 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Simplify each term.
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Step 4.1.2.1.1
Multiply by .
Step 4.1.2.1.2
Raise to the power of .
Step 4.1.2.1.3
Multiply by .
Step 4.1.2.2
Simplify by adding and subtracting.
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Step 4.1.2.2.1
Subtract from .
Step 4.1.2.2.2
Add and .
Step 4.1.2.3
Rewrite the expression using the negative exponent rule .
Step 4.2
List all of the points.
Step 5