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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?
d. For what numbers x, 0 ≤ x ≤ 2π, does f(x) = 0?
e. What are the x-intercepts of f?
Bot GPT 3.5
answered
a. To find the y-intercept of the graph of f, we set x = 0 and evaluate f(0).
f(0) = 4sin(0) = 4(0) = 0
Therefore, the y-intercept of the graph of f is 0.
b. To determine when the graph of f is increasing, we need to find the values of x for which the derivative of f is positive.
f'(x) = 4cos(x)
For the graph of f to increase, f'(x) must be greater than 0.
4cos(x) > 0
cos(x) > 0
In the interval -π ≤ x ≤ π, cos(x) is positive when -π/2 < x < π/2.
So, the graph of f is increasing for -π/2 < x < π/2.
c. The absolute maximum value of f can be found by examining the maximum value of the sine function, which is 1, and multiplying it by the coefficient 4.
Therefore, the absolute maximum value of f is 4.
d. To find the values of x for which f(x) = 0, we need to solve the equation 4sin(x) = 0.
Setting 4sin(x) = 0, we have sin(x) = 0.
In the interval 0 ≤ x ≤ 2π, sin(x) is 0 at x = 0, π, and 2π.
Therefore, for 0 ≤ x ≤ 2π, f(x) = 0 at x = 0, π, and 2π.
e. The x-intercepts of f are the values of x for which f(x) = 0.
From part d, we found that f(x) = 0 at x = 0, π, and 2π.
Therefore, the x-intercepts of f are x = 0, π, and 2π.