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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Find the first derivative.
Step 2.1.1
Differentiate.
Step 2.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2
Evaluate .
Step 2.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.3
Multiply by .
Step 2.1.3
Evaluate .
Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.3
Multiply by .
Step 2.2
Find the second derivative.
Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Evaluate .
Step 2.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.2.3
Multiply by .
Step 2.2.3
Evaluate .
Step 2.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.3
Multiply by .
Step 2.2.4
Differentiate using the Constant Rule.
Step 2.2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.4.2
Add and .
Step 2.3
The second derivative of with respect to is .
Step 3
Step 3.1
Set the second derivative equal to .
Step 3.2
Add to both sides of the equation.
Step 3.3
Divide each term in by and simplify.
Step 3.3.1
Divide each term in by .
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Cancel the common factor of .
Step 3.3.2.1.1
Cancel the common factor.
Step 3.3.2.1.2
Divide by .
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Divide by .
Step 4
Step 4.1
Substitute in to find the value of .
Step 4.1.1
Replace the variable with in the expression.
Step 4.1.2
Simplify the result.
Step 4.1.2.1
Simplify each term.
Step 4.1.2.1.1
One to any power is one.
Step 4.1.2.1.2
One to any power is one.
Step 4.1.2.1.3
Multiply by .
Step 4.1.2.1.4
Multiply by .
Step 4.1.2.2
Simplify by subtracting numbers.
Step 4.1.2.2.1
Subtract from .
Step 4.1.2.2.2
Subtract from .
Step 4.1.2.3
The final answer is .
Step 4.2
The point found by substituting in is . This point can be an inflection point.
Step 5
Split into intervals around the points that could potentially be inflection points.
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
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
Multiply by .
Step 7.2.2
Subtract from .
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 point in this case is .
Step 9