Calculus Examples

Find the Absolute Max and Min over the Interval f(x) = square root of x+5
Step 1
Find the first derivative of the function.
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Step 1.1
Use to rewrite as .
Step 1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Replace all occurrences of with .
Step 1.3
To write as a fraction with a common denominator, multiply by .
Step 1.4
Combine and .
Step 1.5
Combine the numerators over the common denominator.
Step 1.6
Simplify the numerator.
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Step 1.6.1
Multiply by .
Step 1.6.2
Subtract from .
Step 1.7
Combine fractions.
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Step 1.7.1
Move the negative in front of the fraction.
Step 1.7.2
Combine and .
Step 1.7.3
Move to the denominator using the negative exponent rule .
Step 1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.9
Differentiate using the Power Rule which states that is where .
Step 1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.11
Simplify the expression.
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Step 1.11.1
Add and .
Step 1.11.2
Multiply by .
Step 2
Find the second derivative of the function.
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Step 2.1
Differentiate using the Constant Multiple Rule.
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Step 2.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2
Apply basic rules of exponents.
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Step 2.1.2.1
Rewrite as .
Step 2.1.2.2
Multiply the exponents in .
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Step 2.1.2.2.1
Apply the power rule and multiply exponents, .
Step 2.1.2.2.2
Combine and .
Step 2.1.2.2.3
Move the negative in front of the fraction.
Step 2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Replace all occurrences of with .
Step 2.3
To write as a fraction with a common denominator, multiply by .
Step 2.4
Combine and .
Step 2.5
Combine the numerators over the common denominator.
Step 2.6
Simplify the numerator.
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Step 2.6.1
Multiply by .
Step 2.6.2
Subtract from .
Step 2.7
Combine fractions.
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Step 2.7.1
Move the negative in front of the fraction.
Step 2.7.2
Combine and .
Step 2.7.3
Move to the denominator using the negative exponent rule .
Step 2.7.4
Multiply by .
Step 2.7.5
Multiply by .
Step 2.8
By the Sum Rule, the derivative of with respect to is .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Since is constant with respect to , the derivative of with respect to is .
Step 2.11
Simplify the expression.
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Step 2.11.1
Add and .
Step 2.11.2
Multiply by .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
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Step 4.1
Find the first derivative.
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Step 4.1.1
Use to rewrite as .
Step 4.1.2
Differentiate using the chain rule, which states that is where and .
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Step 4.1.2.1
To apply the Chain Rule, set as .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Replace all occurrences of with .
Step 4.1.3
To write as a fraction with a common denominator, multiply by .
Step 4.1.4
Combine and .
Step 4.1.5
Combine the numerators over the common denominator.
Step 4.1.6
Simplify the numerator.
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Step 4.1.6.1
Multiply by .
Step 4.1.6.2
Subtract from .
Step 4.1.7
Combine fractions.
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Step 4.1.7.1
Move the negative in front of the fraction.
Step 4.1.7.2
Combine and .
Step 4.1.7.3
Move to the denominator using the negative exponent rule .
Step 4.1.8
By the Sum Rule, the derivative of with respect to is .
Step 4.1.9
Differentiate using the Power Rule which states that is where .
Step 4.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.11
Simplify the expression.
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Step 4.1.11.1
Add and .
Step 4.1.11.2
Multiply by .
Step 4.2
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
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Step 5.1
Set the first derivative equal to .
Step 5.2
Set the numerator equal to zero.
Step 5.3
Since , there are no solutions.
No solution
No solution
Step 6
Find the values where the derivative is undefined.
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Step 6.1
Convert expressions with fractional exponents to radicals.
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Step 6.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 6.1.2
Anything raised to is the base itself.
Step 6.2
Set the denominator in equal to to find where the expression is undefined.
Step 6.3
Solve for .
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Step 6.3.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 6.3.2
Simplify each side of the equation.
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Step 6.3.2.1
Use to rewrite as .
Step 6.3.2.2
Simplify the left side.
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Step 6.3.2.2.1
Simplify .
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Step 6.3.2.2.1.1
Apply the product rule to .
Step 6.3.2.2.1.2
Raise to the power of .
Step 6.3.2.2.1.3
Multiply the exponents in .
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Step 6.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 6.3.2.2.1.3.2
Cancel the common factor of .
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Step 6.3.2.2.1.3.2.1
Cancel the common factor.
Step 6.3.2.2.1.3.2.2
Rewrite the expression.
Step 6.3.2.2.1.4
Simplify.
Step 6.3.2.2.1.5
Apply the distributive property.
Step 6.3.2.2.1.6
Multiply by .
Step 6.3.2.3
Simplify the right side.
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Step 6.3.2.3.1
Raising to any positive power yields .
Step 6.3.3
Solve for .
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Step 6.3.3.1
Subtract from both sides of the equation.
Step 6.3.3.2
Divide each term in by and simplify.
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Step 6.3.3.2.1
Divide each term in by .
Step 6.3.3.2.2
Simplify the left side.
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Step 6.3.3.2.2.1
Cancel the common factor of .
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Step 6.3.3.2.2.1.1
Cancel the common factor.
Step 6.3.3.2.2.1.2
Divide by .
Step 6.3.3.2.3
Simplify the right side.
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Step 6.3.3.2.3.1
Divide by .
Step 6.4
Set the radicand in less than to find where the expression is undefined.
Step 6.5
Subtract from both sides of the inequality.
Step 6.6
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 7
Critical points to evaluate.
Step 8
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 9
Evaluate the second derivative.
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Step 9.1
Simplify the expression.
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Step 9.1.1
Add and .
Step 9.1.2
Rewrite as .
Step 9.1.3
Apply the power rule and multiply exponents, .
Step 9.2
Cancel the common factor of .
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Step 9.2.1
Cancel the common factor.
Step 9.2.2
Rewrite the expression.
Step 9.3
Simplify the expression.
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Step 9.3.1
Raising to any positive power yields .
Step 9.3.2
Multiply by .
Step 9.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 9.4
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 10
Since the first derivative test failed, there are no local extrema.
No Local Extrema
Step 11