Calculus Examples

Find the Critical Points 12x^2-176x+484
Step 1
Find the first derivative.
Tap for more steps...
Step 1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
Tap for more steps...
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 .
Tap for more steps...
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 1.1.4
Differentiate using the Constant Rule.
Tap for more steps...
Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Add and .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
Tap for more steps...
Step 2.1
Set the first derivative equal to .
Step 2.2
Add to both sides of the equation.
Step 2.3
Divide each term in by and simplify.
Tap for more steps...
Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
Tap for more steps...
Step 2.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
Tap for more steps...
Step 2.3.3.1
Cancel the common factor of and .
Tap for more steps...
Step 2.3.3.1.1
Factor out of .
Step 2.3.3.1.2
Cancel the common factors.
Tap for more steps...
Step 2.3.3.1.2.1
Factor out of .
Step 2.3.3.1.2.2
Cancel the common factor.
Step 2.3.3.1.2.3
Rewrite the expression.
Step 3
Find the values where the derivative is undefined.
Tap for more steps...
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.
Tap for more steps...
Step 4.1
Evaluate at .
Tap for more steps...
Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
Tap for more steps...
Step 4.1.2.1
Simplify each term.
Tap for more steps...
Step 4.1.2.1.1
Apply the product rule to .
Step 4.1.2.1.2
Raise to the power of .
Step 4.1.2.1.3
Raise to the power of .
Step 4.1.2.1.4
Cancel the common factor of .
Tap for more steps...
Step 4.1.2.1.4.1
Factor out of .
Step 4.1.2.1.4.2
Factor out of .
Step 4.1.2.1.4.3
Cancel the common factor.
Step 4.1.2.1.4.4
Rewrite the expression.
Step 4.1.2.1.5
Combine and .
Step 4.1.2.1.6
Multiply by .
Step 4.1.2.1.7
Multiply .
Tap for more steps...
Step 4.1.2.1.7.1
Combine and .
Step 4.1.2.1.7.2
Multiply by .
Step 4.1.2.1.8
Move the negative in front of the fraction.
Step 4.1.2.2
Combine fractions.
Tap for more steps...
Step 4.1.2.2.1
Combine the numerators over the common denominator.
Step 4.1.2.2.2
Simplify the expression.
Tap for more steps...
Step 4.1.2.2.2.1
Subtract from .
Step 4.1.2.2.2.2
Move the negative in front of the fraction.
Step 4.1.2.3
To write as a fraction with a common denominator, multiply by .
Step 4.1.2.4
Combine and .
Step 4.1.2.5
Combine the numerators over the common denominator.
Step 4.1.2.6
Simplify the numerator.
Tap for more steps...
Step 4.1.2.6.1
Multiply by .
Step 4.1.2.6.2
Subtract from .
Step 4.1.2.7
Move the negative in front of the fraction.
Step 4.2
List all of the points.
Step 5