Question in graphing

Below is the graph of some unknown function y = f(x)

Use interval notation to estimate where …

f ‘(x) is positive

f ‘(x) is negative

f ”(x) is positive

f ”(x) is negative

Sketch a graph of what y = f ”(x) might look like.

Answer

First derivative is positive on the intervals the function is increasing and negative on intervals the function is decreasing.

f'(x) > 0 on (-3.5, 0.5) reunited with (2.5, infinity)

f'(x) < 0 on (-infinity, -3.5) reunited with (0.5, 2.5)

at points x =-3.5, x =0.5, x =2.5 the first derivative f'(x) = 0

second derivative is positive where the function holds water (concave), and negative where the function does not hold water (convex).

f”(x) > 0 on (-infinity, -2) reunited with (1.5, +infinity)

f”(x) < 0 on (-2, 1.5)

f”(x) = 0 at points x =-2 and x =1.5

The graph of f”'(x) is below