Suppose the derivative of a function f is f '(x) = (x + 2)4(x − 3)3(x − 5)6. on what interval is f increasing? (enter your answer in interval notation.)

(3, 5) U (5, ∞)

Step-by-step explanation:

The derivative has zeros at -2, 3, and 5. It changes sign only at x=3, being zero or negative to the left of that point, and being zero or positive to the right of that point. The function increases where the derivative is positive (not zero), so on the interval ...

(3, 5) U (5, ∞)

The function f'(x) will touch the x-axis, but not cross, at x=-2 and x=5, where the powers of the factors (x+2) and (x-5) are even. The function will cross the axis where the power of the factor is odd, at x=3.

Each place where the derivative is zero corresponds to a flat spot in the function. (The function is neither increasing nor decreasing there.) The higher the power of the corresponding factor in the derivative, the "flatter" the function is at that point.

step-by-step explanation:

answer 1 : option "c"

answer 2 : option "d"

answer 3 : option "b"

answer 4 : option "d"

answer 5 : option "b"

answer 6 : option "a"

answer 7 : option "c"

answer 8 : option "d"

answer 9 : option "c"

answer 10 : option "d"

answer 11 : option "c"

answer 12 option "b"

answer 13 option "c"

answer 14 option "c"

answer 15 option "d"

answer 16 option "a"

answer 17 option "b"

answer 18 option "c"

answer 19 option "a"

answer 20 option "a"

answer 21 option "c"

d