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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Find the second derivative.
Step 2.1.1
Find the first derivative.
Step 2.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.2
Evaluate .
Step 2.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.1.2.3
Multiply by .
Step 2.1.1.3
Evaluate .
Step 2.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.1.3.3
Multiply by .
Step 2.1.1.4
Differentiate using the Constant Rule.
Step 2.1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.1.4.2
Add and .
Step 2.1.2
Find the second derivative.
Step 2.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2.2
Evaluate .
Step 2.1.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.2.3
Multiply by .
Step 2.1.2.3
Differentiate using the Constant Rule.
Step 2.1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.3.2
Add and .
Step 2.1.3
The second derivative of with respect to is .
Step 2.2
Set the second derivative equal to then solve the equation .
Step 2.2.1
Set the second derivative equal to .
Step 2.2.2
Since , there are no solutions.
No solution
No solution
No solution
Step 3
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.
Interval Notation:
Set-Builder Notation:
Step 4
The graph is concave up because the second derivative is positive.
The graph is concave up
Step 5