# a differential function f has critical numbers at x=4 and x=9. identify all relative extrema of f at each of the critical points if f'(0)=2, f'(6)=-3 and f'(10)=-5.

marizi | High School Teacher | (Level 1) Associate Educator

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According to the first derivative test, the relative extrema can be classified as:

1. Relative minimum if  f'(x) = negative  from the left of x=c and f'(x) = positive to the right of x=c.

2. Relative maximum if  f'(x) = positive  from the left of x=c and f'(x) =negative to the right of x=c.

3. If there is no sign change for f'(x) from the left and right of x=c, then f(x) is not a relative extrema of f.

Solution:

For critical number x=4, we consider

f'(0) =2   which is at the LEFT of x=4 and

f'(6) =-3 which is at the RIGHT of x=4.

Conclusion: x=4 is a relative maximum.

For critical number x=9, we consider

f'(6) =-3   which is at the LEFT of x=4 and

f'(10) =-5 which is at the RIGHT of x=4.

Conclusion: There is no sign change then x=9 is a  NOT a relative extrema.

For critical number x=4, we consider

f'(0) =2   which is at the LEFT of x=4 and

f'(6) =-3 which is at the RIGHT of x=4.

Conclusion: x=4 is a relative maximum.

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