Question
Let f(x) = 4 sin 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 maximum value of f?
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 maximum value of f?
Answers
Answered by
GPT 3.5
a. To find the y-intercept of the graph of f, we need to find f(0). Plugging in x = 0 into the equation f(x) = 4 sin x, we get:
f(0) = 4 sin 0 = 4 * 0 = 0
Therefore, the y-intercept of the graph of f is 0.
b. The graph of f is increasing when the derivative of f with respect to x is positive. Let's find the derivative of f:
f'(x) = 4 cos x
To find the values of x where the graph of f is increasing, we need to find the values of x for which f'(x) > 0. Since cos x is positive in the interval -π ≤ x ≤ π, we have:
f'(x) = 4 cos x > 0
Therefore, the graph of f is increasing for all numbers x in the interval -π ≤ x ≤ π.
c. The absolute maximum value of f occurs at the maximum value of the sine function, which is 1. So, the absolute maximum value of f is:
f(max) = 4 sin(max) = 4 * 1 = 4
f(0) = 4 sin 0 = 4 * 0 = 0
Therefore, the y-intercept of the graph of f is 0.
b. The graph of f is increasing when the derivative of f with respect to x is positive. Let's find the derivative of f:
f'(x) = 4 cos x
To find the values of x where the graph of f is increasing, we need to find the values of x for which f'(x) > 0. Since cos x is positive in the interval -π ≤ x ≤ π, we have:
f'(x) = 4 cos x > 0
Therefore, the graph of f is increasing for all numbers x in the interval -π ≤ x ≤ π.
c. The absolute maximum value of f occurs at the maximum value of the sine function, which is 1. So, the absolute maximum value of f is:
f(max) = 4 sin(max) = 4 * 1 = 4