Question

The derivative of a function is f'(x)=x(x 2)(x-4). from this function, which of the following must be true about f'?

A) f′(x) is always positive.
B) f′(x) has roots at x=0,x=2, and x=4.
C) f′(x) is an increasing function.
D) f′(x) has a local minimum at x=4.

Answer 1

The function f'(x) = x(x-2)(x-4) must have roots at x=0, x=2, and x=4. Without further information, we cannot conclude that it has a local minimum at x=4, is always positive, or is an increasing function.

Step-by-step explanation:

The derivative of the function is f'(x) = x(x-2)(x-4). From this, we can determine several characteristics about the function f'. First, f'(x) has roots at x=0, x=2, and x=4, which means these are the x-values where the function will cross the x-axis. So option B is correct.

Moreover, to determine if f'(x) has a local minimum at x=4, we would need to analyze the signs of the function around these roots as well as the second derivative, which is not provided. Hence, we cannot assert that option D is true without further information. Since the signs of the function f'(x) before and after the roots determine the function's behavior as being positive or negative, option A is not necessarily true as f'(x) can change signs around the roots. Lastly, without additional information concerning the derivatives beyond the first, we cannot conclude that f'(x) is an increasing function (option C).