Let f(x) = 4 cos x.
Complete parts (a) through (e) below
a. What is the y-intercept of the graph of f?
b. For what numbers x, -π ≤ x ≤ π, is the graph of f increasing?
c. What is the absolute minimum value of f?
d. For what numbers x, 0 ≤ x ≤ 2π, does f(x) = 0?
e. What are the x-intercepts of f?
a. The y-intercept of a graph is the value of the function when x = 0. Therefore, to find the y-intercept of f(x) = 4 cos x, we substitute x = 0 into the function:
f(0) = 4 cos 0 = 4(1) = 4.
So, the y-intercept of the graph of f is 4.
b. To determine when the graph of f is increasing, we need to identify the values of x where the derivative of f is positive. The derivative of f(x) = 4 cos x is:
f'(x) = -4 sin x.
The graph of f will be increasing when f'(x) > 0. Given the interval -π ≤ x ≤ π, sin x is positive for x > 0 (0 < x < π) and negative for x < 0 (-π < x < 0). Therefore, the graph of f is increasing for 0 < x < π.
c. The absolute minimum value of f occurs when the cosine function has its lowest value, which is -1. When x = π, cos π = -1. Substituting this into the function:
f(π) = 4 cos π = 4(-1) = -4.
So, the absolute minimum value of f is -4.
d. To find the values of x for which f(x) = 0, we need to solve the equation 4 cos x = 0. We know that cos x = 0 when x = π/2 and x = 3π/2. Since the given interval is 0 ≤ x ≤ 2π, these values of x fall within this interval.
Therefore, f(x) = 0 at x = π/2 and x = 3π/2.
e. The x-intercepts of f occur when y = 0. Therefore, we need to solve the equation 4 cos x = 0. We know that cos x = 0 when x = π/2 and x = 3π/2. These are the x-intercepts of f.
Therefore, the x-intercepts of f are x = π/2 and x = 3π/2.