Calculus Examples

Find the Local Maxima and Minima f(x)=x^2 natural log of x
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the Product Rule which states that is where and .
Step 1.2
The derivative of with respect to is .
Step 1.3
Differentiate using the Power Rule.
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Step 1.3.1
Combine and .
Step 1.3.2
Cancel the common factor of and .
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Step 1.3.2.1
Factor out of .
Step 1.3.2.2
Cancel the common factors.
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Step 1.3.2.2.1
Raise to the power of .
Step 1.3.2.2.2
Factor out of .
Step 1.3.2.2.3
Cancel the common factor.
Step 1.3.2.2.4
Rewrite the expression.
Step 1.3.2.2.5
Divide by .
Step 1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.3.4
Reorder terms.
Step 2
Find the second derivative of the function.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Product Rule which states that is where and .
Step 2.2.3
The derivative of with respect to is .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Combine and .
Step 2.2.6
Cancel the common factor of .
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Step 2.2.6.1
Cancel the common factor.
Step 2.2.6.2
Rewrite the expression.
Step 2.2.7
Multiply by .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Combine terms.
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Step 2.4.2.1
Multiply by .
Step 2.4.2.2
Add and .
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
Differentiate using the Product Rule which states that is where and .
Step 4.1.2
The derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule.
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Step 4.1.3.1
Combine and .
Step 4.1.3.2
Cancel the common factor of and .
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Step 4.1.3.2.1
Factor out of .
Step 4.1.3.2.2
Cancel the common factors.
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Step 4.1.3.2.2.1
Raise to the power of .
Step 4.1.3.2.2.2
Factor out of .
Step 4.1.3.2.2.3
Cancel the common factor.
Step 4.1.3.2.2.4
Rewrite the expression.
Step 4.1.3.2.2.5
Divide by .
Step 4.1.3.3
Differentiate using the Power Rule which states that is where .
Step 4.1.3.4
Reorder terms.
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
Subtract from both sides of the equation.
Step 5.3
Divide each term in by and simplify.
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Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Cancel the common factor of .
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Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Rewrite the expression.
Step 5.3.2.2
Cancel the common factor of .
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Step 5.3.2.2.1
Cancel the common factor.
Step 5.3.2.2.2
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Cancel the common factor of .
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Step 5.3.3.1.1
Cancel the common factor.
Step 5.3.3.1.2
Rewrite the expression.
Step 5.3.3.2
Move the negative in front of the fraction.
Step 5.4
To solve for , rewrite the equation using properties of logarithms.
Step 5.5
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 5.6
Solve for .
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Step 5.6.1
Rewrite the equation as .
Step 5.6.2
Rewrite the expression using the negative exponent rule .
Step 6
Find the values where the derivative is undefined.
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Step 6.1
Set the argument in less than or equal to to find where the expression is undefined.
Step 6.2
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 each term.
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Step 9.1.1
Move to the numerator using the negative exponent rule .
Step 9.1.2
Expand by moving outside the logarithm.
Step 9.1.3
The natural logarithm of is .
Step 9.1.4
Multiply by .
Step 9.1.5
Cancel the common factor of .
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Step 9.1.5.1
Move the leading negative in into the numerator.
Step 9.1.5.2
Cancel the common factor.
Step 9.1.5.3
Rewrite the expression.
Step 9.2
Add and .
Step 10
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 11
Find the y-value when .
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Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
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Step 11.2.1
Simplify the expression.
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Step 11.2.1.1
Apply the product rule to .
Step 11.2.1.2
One to any power is one.
Step 11.2.2
Simplify the denominator.
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Step 11.2.2.1
Multiply the exponents in .
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Step 11.2.2.1.1
Apply the power rule and multiply exponents, .
Step 11.2.2.1.2
Cancel the common factor of .
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Step 11.2.2.1.2.1
Cancel the common factor.
Step 11.2.2.1.2.2
Rewrite the expression.
Step 11.2.2.2
Simplify.
Step 11.2.3
Move to the numerator using the negative exponent rule .
Step 11.2.4
Expand by moving outside the logarithm.
Step 11.2.5
The natural logarithm of is .
Step 11.2.6
Multiply by .
Step 11.2.7
Multiply by .
Step 11.2.8
Move to the left of .
Step 11.2.9
The final answer is .
Step 12
These are the local extrema for .
is a local minima
Step 13