Calculus Examples

Find the Local Maxima and Minima f(x)=7x-2x natural log of x
Step 1
Find the first derivative of the function.
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Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
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Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Product Rule which states that is where and .
Step 1.3.3
The derivative of with respect to is .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Combine and .
Step 1.3.6
Cancel the common factor of .
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Step 1.3.6.1
Cancel the common factor.
Step 1.3.6.2
Rewrite the expression.
Step 1.3.7
Multiply by .
Step 1.4
Simplify.
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Step 1.4.1
Apply the distributive property.
Step 1.4.2
Combine terms.
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Step 1.4.2.1
Multiply by .
Step 1.4.2.2
Subtract from .
Step 2
Find the second derivative of the function.
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Step 2.1
Differentiate.
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Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Since is constant with respect to , 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
The derivative of with respect to is .
Step 2.2.3
Combine and .
Step 2.2.4
Move the negative in front of the fraction.
Step 2.3
Subtract from .
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
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.1.3
Evaluate .
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Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Product Rule which states that is where and .
Step 4.1.3.3
The derivative of with respect to is .
Step 4.1.3.4
Differentiate using the Power Rule which states that is where .
Step 4.1.3.5
Combine and .
Step 4.1.3.6
Cancel the common factor of .
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Step 4.1.3.6.1
Cancel the common factor.
Step 4.1.3.6.2
Rewrite the expression.
Step 4.1.3.7
Multiply by .
Step 4.1.4
Simplify.
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Step 4.1.4.1
Apply the distributive property.
Step 4.1.4.2
Combine terms.
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Step 4.1.4.2.1
Multiply by .
Step 4.1.4.2.2
Subtract from .
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
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Dividing two negative values results in a positive value.
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
Rewrite the equation as .
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
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 10
Find the y-value when .
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Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
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Step 10.2.1
Simplify each term.
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Step 10.2.1.1
Use logarithm rules to move out of the exponent.
Step 10.2.1.2
The natural logarithm of is .
Step 10.2.1.3
Multiply by .
Step 10.2.1.4
Cancel the common factor of .
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Step 10.2.1.4.1
Factor out of .
Step 10.2.1.4.2
Cancel the common factor.
Step 10.2.1.4.3
Rewrite the expression.
Step 10.2.1.5
Multiply by .
Step 10.2.2
Subtract from .
Step 10.2.3
The final answer is .
Step 11
These are the local extrema for .
is a local maxima
Step 12