Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=- square root of x+2 ; 0<=x<=5
;
Step 1
Find the critical points.
Tap for more steps...
Step 1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1.1
Differentiate using the Constant Multiple Rule.
Tap for more steps...
Step 1.1.1.1.1
Use to rewrite as .
Step 1.1.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Replace all occurrences of with .
Step 1.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.4
Combine and .
Step 1.1.1.5
Combine the numerators over the common denominator.
Step 1.1.1.6
Simplify the numerator.
Tap for more steps...
Step 1.1.1.6.1
Multiply by .
Step 1.1.1.6.2
Subtract from .
Step 1.1.1.7
Combine fractions.
Tap for more steps...
Step 1.1.1.7.1
Move the negative in front of the fraction.
Step 1.1.1.7.2
Combine and .
Step 1.1.1.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.9
Differentiate using the Power Rule which states that is where .
Step 1.1.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.11
Simplify the expression.
Tap for more steps...
Step 1.1.1.11.1
Add and .
Step 1.1.1.11.2
Multiply by .
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
Tap for more steps...
Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Since , there are no solutions.
No solution
No solution
Step 1.3
Find the values where the derivative is undefined.
Tap for more steps...
Step 1.3.1
Convert expressions with fractional exponents to radicals.
Tap for more steps...
Step 1.3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 1.3.1.2
Anything raised to is the base itself.
Step 1.3.2
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.3
Solve for .
Tap for more steps...
Step 1.3.3.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 1.3.3.2
Simplify each side of the equation.
Tap for more steps...
Step 1.3.3.2.1
Use to rewrite as .
Step 1.3.3.2.2
Simplify the left side.
Tap for more steps...
Step 1.3.3.2.2.1
Simplify .
Tap for more steps...
Step 1.3.3.2.2.1.1
Apply the product rule to .
Step 1.3.3.2.2.1.2
Raise to the power of .
Step 1.3.3.2.2.1.3
Multiply the exponents in .
Tap for more steps...
Step 1.3.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 1.3.3.2.2.1.3.2
Cancel the common factor of .
Tap for more steps...
Step 1.3.3.2.2.1.3.2.1
Cancel the common factor.
Step 1.3.3.2.2.1.3.2.2
Rewrite the expression.
Step 1.3.3.2.2.1.4
Simplify.
Step 1.3.3.2.2.1.5
Apply the distributive property.
Step 1.3.3.2.2.1.6
Multiply by .
Step 1.3.3.2.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.2.3.1
Raising to any positive power yields .
Step 1.3.3.3
Solve for .
Tap for more steps...
Step 1.3.3.3.1
Subtract from both sides of the equation.
Step 1.3.3.3.2
Divide each term in by and simplify.
Tap for more steps...
Step 1.3.3.3.2.1
Divide each term in by .
Step 1.3.3.3.2.2
Simplify the left side.
Tap for more steps...
Step 1.3.3.3.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.3.3.3.2.2.1.1
Cancel the common factor.
Step 1.3.3.3.2.2.1.2
Divide by .
Step 1.3.3.3.2.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.3.2.3.1
Divide by .
Step 1.3.4
Set the radicand in less than to find where the expression is undefined.
Step 1.3.5
Subtract from both sides of the inequality.
Step 1.3.6
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
Step 1.4
Evaluate at each value where the derivative is or undefined.
Tap for more steps...
Step 1.4.1
Evaluate at .
Tap for more steps...
Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
Tap for more steps...
Step 1.4.1.2.1
Add and .
Step 1.4.1.2.2
Rewrite as .
Step 1.4.1.2.3
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.1.2.4
Multiply by .
Step 1.4.2
List all of the points.
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
Tap for more steps...
Step 3.1
Evaluate at .
Tap for more steps...
Step 3.1.1
Substitute for .
Step 3.1.2
Add and .
Step 3.2
Evaluate at .
Tap for more steps...
Step 3.2.1
Substitute for .
Step 3.2.2
Add and .
Step 3.3
List all of the points.
Step 4
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
Step 5