Choose the one alternative that best completes the statements or answers the que

Question

Choose the one alternative that best completes the statements or answers the question. of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. Never decreasing increasing on (- x, 5) (5, -) Never increasing; decreasing on (-x, -5) (-5, x) Never decreasing; increasing on (-w, -5) (-5, x) Decreasing on (-x, -5); increasing on (-5, x) (x) = x^1/3 (x - 7) Increasing on (0, x) Decreasing on (0, 7); increasing on (- x, 0) (7, x) Decreasing on (- x, 0) (7, x); increasing on (0, 7) Decreasing on (0, 7); increasing on (7, x)

Explanation / Answer

1) The since (x+5) and e^x both are always positive, the derivative f’(x)=(x+5)^2e^x is always positive for all values of x. Also, x=-5 is the critical point of the function f(x), so the function f(x) is never decreasing and it is increasing on (-, -5)(-5, ).

So, option (A) is correct.

2) Since, f’(x)=x^(1/3)(x-7), the critical points are x=0 and x=7

Now, f’(x) < 0 in (0, 7) and f’(x)>0 in (-, 0)(7, ). Thus, the function is decreasing on (0, 7) and increasing on the interval (-, 0)(7, ).

So, option (B) is correct.

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