Question

Answer the following questions about the function whose derivative is f'(x) = x^(-1/5)(x - 4): (a) What are the critical points of f? (b) On what open intervals is f increasing?

Answer 1

The critical points of the function with the given derivative f'(x) = x^(-1/5)(x - 4) are x = 0 and x = 4. The function f is increasing on the interval (4, ∞).

Step-by-step explanation:

The student is asking several different questions regarding calculus, probability, and physics, which are all related to the subject of Mathematics at the college level. For the question about the function with the derivative f'(x) = x^{(-1/5)}(x - 4), we are asked to:

• (a) Determine the critical points of f.
• (b) Identify the open intervals where f is increasing.

Critical points occur where the derivative is equal to zero or undefined. In this case, f'(x) is zero when x = 4 and undefined when x = 0. Therefore, the critical points are x = 0 and x = 4.

To determine where the function f is increasing, we examine the sign of f'(x) on either side of the critical points. For x > 4, the derivative is positive, implying that f is increasing on the interval (4, ∞). For 0 < x < 4, the derivative is negative, meaning f is decreasing on this interval. Therefore, f is increasing on the open interval (4, ∞).