Finding the image of a function based off the graph.

Question

https://ibb.co/yXTX407 (the graph relating to the question)

The question states: "The graph shows the curve of the function f(x) where Df = [-2,2]. Use the photo to work out the image of f"

(The question is then followed by

b) solve the equation f(x) = 0

c) solve the equation f'(x) = 0

d) solve the equation f''(x) = 0

e) at what interval is the function curved down?

These are just extras but I could also do with some help with these)

Answer 1

To find the image of f, we need to look at the vertical axis of the graph, which represents the range of the function. From the graph, we can see that the lowest point on the curve is approximately -3 and the highest point is approximately 2. Therefore, the image of f is [-3,2].

(b) To solve the equation f(x) = 0, we need to find the x-values where the curve intersects the x-axis. From the graph, we can see that there are two such points, one at x ≈ -1.5 and one at x ≈ 1. Therefore, the solutions to f(x) = 0 are approximately x = -1.5 and x = 1.

(c) To solve the equation f'(x) = 0, we need to find the x-values where the slope of the curve is zero. From the graph, we can see that there are three such points, one at x ≈ -1, one at x ≈ 0, and one at x ≈ 1.5. Therefore, the solutions to f'(x) = 0 are approximately x = -1, x = 0, and x = 1.5.

(d) To solve the equation f''(x) = 0, we need to find the x-values where the curvature of the curve changes. From the graph, we can see that there is only one such point, at x ≈ 0. Therefore, the solution to f''(x) = 0 is approximately x = 0.

(e) The function is curved down when the second derivative f''(x) is negative. From part (d), we know that the curvature changes at x ≈ 0. From the graph, we can see that the curve is curved down to the left of x ≈ 0, and curved up to the right of x ≈ 0. Therefore, the interval where the function is curved down is approximately [-2,0].