How do use the first derivative test to determine the local extrema #f(x)= x^3 - x^2 - 40x + 8#?

Question

Isabella Ackerman · Accepted Answer

This function has a local maximum value of #2516/27# at #x=-10/3# and a local minimum value of #-104# at #x=4#.

The first derivative is #f'(x)=3x^2-2x-40#. This can be factored as #f'(x)=3x^2-2x-40=(3x+10)(x-4)#. Setting this equal to zero and solving for #x# gives two critical points for #f# at #x=-10/3# and #x=4#. You can check that #f'# changes sign from positive to negative as #x# increases through #x=-10/3# and negative to positive as #x# increases through #x=4# (note that #f'# is a continuous function and that, for instance, #f'(-4)=48+8-40=16>0#, #f'(0)=-40<0#, and #f'(5)=75-10-40=25>0#). Therefore, the First Derivative Test implies that #f# has a local maximum at #x=-10/3# and a local minimum at #x=4#. The local maximum value (output) is #f(-10/3)=-1000/27-100/9+400/3+8=2516/27# and the local minimum value (output) is #f(4)=64-16-160+8=-104#.

William Abdalla · Answer

undefined To use the first derivative test to determine the local extrema of $ f(x) = x^3 - x^2 - 40x + 8 $, follow these steps:

1. Find the first derivative of the function, $ f'(x) $.
   $ f'(x) = 3x^2 - 2x - 40 $.

2. Find the critical points by setting $ f'(x) $ equal to zero and solving for $ x $.
   $ 3x^2 - 2x - 40 = 0 $.
   You can solve this quadratic equation to find the critical points.

3. Once you find the critical points, test the intervals around them to determine whether the function is increasing or decreasing.
   - Choose a test point in each interval and plug it into $ f'(x) $.
   - If $ f'(x) > 0 $, the function is increasing.
   - If $ f'(x) < 0 $, the function is decreasing.

4. Analyze the signs of $ f'(x) $ to determine the nature of the local extrema at the critical points.
   - If $ f'(x) $ changes from positive to negative at a critical point, it indicates a local maximum.
   - If $ f'(x) $ changes from negative to positive at a critical point, it indicates a local minimum.
   - If $ f'(x) $ does not change sign at a critical point, the test is inconclusive.

5. Determine the values of $ f(x) $ at the critical points to confirm whether they are local maxima or minima.

Applying these steps will help you use the first derivative test to determine the local extrema of the function $ f(x) = x^3 - x^2 - 40x + 8 $.