Question

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.
a. 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.
b. 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.
c. f is decreasing on the interval (-5,4).
d. f is increasing on the interval (4,∞).

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