# Find the Local Maxima and Minima y=x+25/x Rewrite the equation as a function of .
Find the first derivative of the function.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Multiply by .
Rewrite the expression using the negative exponent rule .
Simplify.
Combine terms.
Combine and .
Move the negative in front of the fraction.
Reorder terms.
Find the second derivative of the function.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the Power Rule which states that is where .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Subtract from .
Multiply by .
Since is constant with respect to , the derivative of with respect to is .
Rewrite the expression using the negative exponent rule .
Combine terms.
Combine and .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Subtract from both sides of the equation.
Solve for .
Multiply each term by and simplify.
Multiply each term in by .
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Rewrite the equation as .
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
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.
Evaluate the second derivative.
Raise to the power of .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
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
Find the y-value when .
Replace the variable with in the expression.
Simplify the result.
Divide by .
The final answer is .
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.
Evaluate the second derivative.
Raise to the power of .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
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
Find the y-value when .
Replace the variable with in the expression.
Simplify the result.
Divide by .
Subtract from .
The final answer is .
These are the local extrema for .
is a local minima
is a local maxima
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