Calculus Examples

Find the Critical Points f(x)=x^4-2x^3
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
Differentiate.
Tap for more steps...
Step 1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
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 1.2
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
Factor out of .
Tap for more steps...
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 and solve for .
Tap for more steps...
Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
Tap for more steps...
Step 2.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.4.2.2
Simplify .
Tap for more steps...
Step 2.4.2.2.1
Rewrite as .
Step 2.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.4.2.2.3
Plus or minus is .
Step 2.5
Set equal to and solve for .
Tap for more steps...
Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
Tap for more steps...
Step 2.5.2.1
Add to both sides of the equation.
Step 2.5.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.5.2.2.1
Divide each term in by .
Step 2.5.2.2.2
Simplify the left side.
Tap for more steps...
Step 2.5.2.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.2.1.2
Divide by .
Step 2.6
The final solution is all the values that make true.
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
Raising to any positive power yields .
Step 4.1.2.1.2
Raising to any positive power yields .
Step 4.1.2.1.3
Multiply by .
Step 4.1.2.2
Add and .
Step 4.2
Evaluate at .
Tap for more steps...
Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
Tap for more steps...
Step 4.2.2.1
Simplify each term.
Tap for more steps...
Step 4.2.2.1.1
Apply the product rule to .
Step 4.2.2.1.2
Raise to the power of .
Step 4.2.2.1.3
Raise to the power of .
Step 4.2.2.1.4
Apply the product rule to .
Step 4.2.2.1.5
Raise to the power of .
Step 4.2.2.1.6
Raise to the power of .
Step 4.2.2.1.7
Cancel the common factor of .
Tap for more steps...
Step 4.2.2.1.7.1
Factor out of .
Step 4.2.2.1.7.2
Factor out of .
Step 4.2.2.1.7.3
Cancel the common factor.
Step 4.2.2.1.7.4
Rewrite the expression.
Step 4.2.2.1.8
Rewrite as .
Step 4.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 4.2.2.3.1
Multiply by .
Step 4.2.2.3.2
Multiply by .
Step 4.2.2.4
Combine the numerators over the common denominator.
Step 4.2.2.5
Simplify the numerator.
Tap for more steps...
Step 4.2.2.5.1
Multiply by .
Step 4.2.2.5.2
Subtract from .
Step 4.2.2.6
Move the negative in front of the fraction.
Step 4.3
List all of the points.
Step 5