Question

let f be a function with first derivative given by f′(x)=(x 1)(x−2)(x−3) . at what values of x does f have a relative maximum?

Answer 1

The values of x where the function f has a relative maximum are x = 1 and x = 3.

Step-by-step explanation:

To find the values of x where the function f has a relative maximum, we need to find the critical points of f. A critical point occurs where the derivative of f equals 0 or does not exist. In this case, the derivative is f'(x) = (x-1)(x-2)(x-3). Setting this equal to 0, we find x = 1, x = 2, and x = 3 as critical points. Since the function is increasing on the interval (-∞, 1) and decreasing on the interval (1, ∞), and also increasing on the interval (2, 3) and decreasing on the interval (3, ∞), the relative maxima occur at x = 1 and x = 3.

Answer 2

To find the relative maximum of the function f, we analyze the first derivative f'(x) which is (x - 1)(x - 2)(x - 3). The critical points are x = 1, 2, and 3. After using the first derivative test, we conclude that f has a relative maximum at x = 2.

Step-by-step explanation:

To determine the values of x at which the function f has a relative maximum, we need to analyze the first derivative f'(x) given by f'(x) = (x - 1)(x - 2)(x - 3). A relative maximum occurs where the first derivative changes from positive to negative.

First, we need to find the critical points by setting the first derivative equal to zero and solving for x:

• f'(x) = 0
• (x - 1)(x - 2)(x - 3) = 0
• x = 1, 2, 3

These critical points are where the function's slope is zero, which could indicate a relative maximum, minimum, or saddle point. To determine the nature of these critical points, we use the first derivative test by evaluating the derivative on intervals around our critical points: x < 1, 1 < x < 2, 2 < x < 3, and x > 3.

After analyzing the sign of the first derivative on each interval, we find:

• For x < 1, f'(x) is negative, indicating a decreasing function.
• For 1 < x < 2, f'(x) is positive, indicating an increasing function.
• For 2 < x < 3, f'(x) is negative, indicating a decreasing function.
• For x > 3, f'(x) is positive, indicating an increasing function.

Therefore, the function f has a relative maximum at x = 2, because the derivative changes from positive to negative as it passes through x = 2. We can disregard x = 1 and x = 3 since the first changes from negative to positive (relative minimum) and the latter from negative to positive (indicating a further increasing behavior or potential relative minimum).