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

Accepted Solution

Answer: (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.