How do you find local maximum value of f using the first and second derivative tests: #f(x) = x / (x^2 + 9)#?

Question

How do you find local maximum value of f using the first and second derivative tests:  #f(x) = x / (x^2 + 9)#?

Emily Ammerman · Accepted Answer

The first derivative yields a critical point at #x=3# i.e.
#(df(x))/(dx)=0, " for " x_(1,2)=+-3#
The second derivative #(d^2f(x))/(dx)^2|_(x=3)=-0.01852#

Since #(d^2f(x))/(dx)^2|_(x=3)<0# we have local maximum

Given: #f(x) = x / (x^2 + 9)# Required: Local Maxima Solution Strategy: 1) Take the 1st derivative to find the critical point, #c#, by solving #f'(x)=0#. The solution is the critical point 2) Take the 2nd derivative and evaluate it at #c#: 1) Use quotient rule to differentiate f(x) #(df(x))/(dx)=(x^2+9 - 2x^2)/(x^2+9)^2=(9-x^2)/(x^2+9)^2 # Now set #f'(x)=0# and solve #0=(9-x^2)/(x^2+9)^2# so the critical points are #x_(1,2)=+-3# 2) #(d^2f(x))/dx^2=(-2x(x^2+9)^2- (4x(x^2+9))(9-x^2))/(x^2+9)^4 # # = 2x(x^2-27)/(x^2+9)^3|_(x=3)=-0.01852 # Thus we have a maximum

Samantha Adkison · Answer

undefined To find the local maximum value of $ f(x) = \frac{x}{x^2 + 9} $ using the first and second derivative tests:

1. Find the first derivative, $ f'(x) $:
$$ f(x) = \frac{x}{x^2 + 9} $$
$$ f'(x) = \frac{(x^2 + 9)(1) - (x)(2x)}{(x^2 + 9)^2} $$
$$ f'(x) = \frac{x^2 + 9 - 2x^2}{(x^2 + 9)^2} $$
$$ f'(x) = \frac{9 - x^2}{(x^2 + 9)^2} $$

2. Find the critical points by setting $ f'(x) = 0 $:
$$ \frac{9 - x^2}{(x^2 + 9)^2} = 0 $$
$$ 9 - x^2 = 0 $$
$$ x^2 = 9 $$
$$ x = \pm 3 $$

3. Find the second derivative, $ f''(x) $:
$$ f'(x) = \frac{9 - x^2}{(x^2 + 9)^2} $$
$$ f''(x) = \frac{d}{dx} \left( \frac{9 - x^2}{(x^2 + 9)^2} \right) $$

To simplify this expression, I suggest you to use a Computer Algebra System or a calculator.

4. Evaluate $ f''(x) $ at the critical points $ x = -3 $ and $ x = 3 $.

5. Determine the nature of the critical points:
   - If $ f''(x) > 0 $, it's a local minimum.
   - If $ f''(x) < 0 $, it's a local maximum.
   - If $ f''(x) = 0 $, the test is inconclusive.

6. Calculate $ f(x) $ at the critical points and compare to find the local maximum value.