suppose f(x) is continuous on (-∞,∞) and f has critical points at x = -5 and x = 4. if we know f(-6) < 0…

suppose f(x) is continuous on (-∞,∞) and f has critical points at x = -5 and x = 4. if we know f(-6) < 0, f(0) > 0, and f(5) < 0, determine whether each of the following statements is true or false.\na. f has a relative minimum at x = -5 because the function is decreasing to the left side of x = -5 and increasing on the right side of x = -5.\nb. f has a relative maximum at x = 4 because f is positive on the left side of x = 4 and negative on the right side of x = 4.\nc. f is decreasing on the interval (-5,4).\nd. f is increasing on the interval (4,∞).

suppose f(x) is continuous on (-∞,∞) and f has critical points at x = -5 and x = 4. if we know f(-6) < 0, f(0) > 0, and f(5) < 0, determine whether each of the following statements is true or false.\na. f has a relative minimum at x = -5 because the function is decreasing to the left side of x = -5 and increasing on the right side of x = -5.\nb. f has a relative maximum at x = 4 because f is positive on the left side of x = 4 and negative on the right side of x = 4.\nc. f is decreasing on the interval (-5,4).\nd. f is increasing on the interval (4,∞).

Answer

Explanation:

Step1: Analyze relative - minimum at x = - 5

Given (f'(-6)<0) and (f'(0)>0). Since (f(x)) is continuous and (f') changes sign from negative (left - hand side of (x = - 5)) to positive (right - hand side of (x=-5)), by the first - derivative test, (f) has a relative minimum at (x = - 5). So the statement a is True.

Step2: Analyze relative - maximum at x = 4

Given (f'(0)>0) and (f'(5)<0). Since (f(x)) is continuous and (f') changes sign from positive (left - hand side of (x = 4)) to negative (right - hand side of (x = 4)), by the first - derivative test, (f) has a relative maximum at (x = 4). So the statement b is True.

Step3: Analyze the interval (-5,4)

Since (f'(0)>0) and (0\in(-5,4)), the function (f(x)) is increasing on some sub - interval of ((-5,4)). So the statement c is False.

Step4: Analyze the interval ((4,\infty))

Since (f'(5)<0) and (5\in(4,\infty)), the function (f(x)) is decreasing on some sub - interval of ((4,\infty)). So the statement d is False.

Answer:

a. True b. True c. False d. False