Find the absolute extrema of the function, if they exist, over the indicated int

Question

Find the absolute extrema of the function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. If no interval is specified, use the real numbers, (- oo,0o) f(x)5x34x2-7x-1;(0,o) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum is at x·1 and the absolute minimum is [ at x OB. The absolute minimum is atx=L and there is no absolute maximum O C. The absolute maximum isat xand there is no absolute minimum. 0 D. (Use a comma to separate answers as needed.) (Use a comma to separate answers as needed.) (Use a comma to separate answers as needed.) There is no absolute maximum and no absolute minimum.

Explanation / Answer

f(x)= -5x3+4x2+7x-1

f'(x) = -15x2+8x +7

Critical points, f'(x)=0

15x2-8x-7=0

(x-1)(15x+7)=0

x=1, x=-7/15

At x=1, f(x) = -5+4+7-1 = 5

At x=-7/15, f(x) = -1949/675

The absolute maximum is at x=1 and absolute minimum is at x=-7/15

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