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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Multiply by .
Step 1.1.3
Evaluate .
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.2
Find the second derivative.
Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Evaluate .
Step 1.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.3
Multiply by .
Step 1.2.3
Evaluate .
Step 1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.3.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3.3
Multiply by .
Step 1.3
The second derivative of with respect to is .
Step 2
Step 2.1
Set the second derivative equal to .
Step 2.2
Factor out of .
Step 2.2.1
Factor out of .
Step 2.2.2
Factor out of .
Step 2.2.3
Factor out of .
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to .
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
Step 2.5.2.1
Subtract from both sides of the equation.
Step 2.5.2.2
Divide each term in by and simplify.
Step 2.5.2.2.1
Divide each term in by .
Step 2.5.2.2.2
Simplify the left side.
Step 2.5.2.2.2.1
Cancel the common factor of .
Step 2.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.2.1.2
Divide by .
Step 2.5.2.2.3
Simplify the right side.
Step 2.5.2.2.3.1
Move the negative in front of the fraction.
Step 2.6
The final solution is all the values that make true.
Step 3
Step 3.1
Substitute in to find the value of .
Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
Step 3.1.2.1
Simplify each term.
Step 3.1.2.1.1
Raising to any positive power yields .
Step 3.1.2.1.2
Multiply by .
Step 3.1.2.1.3
Raising to any positive power yields .
Step 3.1.2.1.4
Multiply by .
Step 3.1.2.2
Add and .
Step 3.1.2.3
The final answer is .
Step 3.2
The point found by substituting in is . This point can be an inflection point.
Step 3.3
Substitute in to find the value of .
Step 3.3.1
Replace the variable with in the expression.
Step 3.3.2
Simplify the result.
Step 3.3.2.1
Simplify each term.
Step 3.3.2.1.1
Use the power rule to distribute the exponent.
Step 3.3.2.1.1.1
Apply the product rule to .
Step 3.3.2.1.1.2
Apply the product rule to .
Step 3.3.2.1.2
Raise to the power of .
Step 3.3.2.1.3
Multiply by .
Step 3.3.2.1.4
Raise to the power of .
Step 3.3.2.1.5
Raise to the power of .
Step 3.3.2.1.6
Cancel the common factor of .
Step 3.3.2.1.6.1
Factor out of .
Step 3.3.2.1.6.2
Cancel the common factor.
Step 3.3.2.1.6.3
Rewrite the expression.
Step 3.3.2.1.7
Use the power rule to distribute the exponent.
Step 3.3.2.1.7.1
Apply the product rule to .
Step 3.3.2.1.7.2
Apply the product rule to .
Step 3.3.2.1.8
Raise to the power of .
Step 3.3.2.1.9
Raise to the power of .
Step 3.3.2.1.10
Raise to the power of .
Step 3.3.2.1.11
Multiply .
Step 3.3.2.1.11.1
Multiply by .
Step 3.3.2.1.11.2
Combine and .
Step 3.3.2.1.11.3
Multiply by .
Step 3.3.2.1.12
Move the negative in front of the fraction.
Step 3.3.2.2
Combine fractions.
Step 3.3.2.2.1
Combine the numerators over the common denominator.
Step 3.3.2.2.2
Simplify the expression.
Step 3.3.2.2.2.1
Subtract from .
Step 3.3.2.2.2.2
Move the negative in front of the fraction.
Step 3.3.2.3
The final answer is .
Step 3.4
The point found by substituting in is . This point can be an inflection point.
Step 3.5
Determine the points that could be inflection points.
Step 4
Split into intervals around the points that could potentially be inflection points.
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.2
Subtract from .
Step 5.2.3
The final answer is .
Step 5.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Raise to the power of .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Multiply by .
Step 6.2.2
Subtract from .
Step 6.2.3
The final answer is .
Step 6.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 7
Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
Step 7.2.1
Simplify each term.
Step 7.2.1.1
Raise to the power of .
Step 7.2.1.2
Multiply by .
Step 7.2.1.3
Multiply by .
Step 7.2.2
Add and .
Step 7.2.3
The final answer is .
Step 7.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 8
An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are .
Step 9