Question

The figure above shows the graph of f ', the derivative of a twice-differentiable function f, on the interval

[–3, 4]. Answer the following questions based on the graph of f ′.
a) Find all x-coordinates at which f has critical values and identify them as a relative minimum,
relative maximum, or neither. Justify your answer.
b) On what open intervals contained in [–3, 4] is the graph of f both concave up and decreasing?
Justify your answer.
c) Find the x-coordinates of all points of inflection on the graph of f

Answer 1

Explanation:

a) The x-coordinates at which f has critical values can be found by looking at the x-coordinates of the points where the graph of f' changes from increasing to decreasing or vice versa. These are the points where the graph of f' crosses the x-axis.

Based on the graph of f', we can see that the critical points occur at x = -2, -1, and 2. To determine if these are relative minimum, relative maximum, or neither, we need to examine the second derivative of f. If the second derivative is positive at a critical point, then the point is a relative minimum. If the second derivative is negative at a critical point, then the point is a relative maximum. If the second derivative is zero at a critical point, then it's neither.

b) The graph of f is concave up when the second derivative is positive, and the graph of f is decreasing when the first derivative is negative. Based on the graph of f', we can see that the graph of f is concave up on the open intervals (-3,-2) and (1,2) and decreasing on the open intervals (-infinity,-3) and (-2,-1) U (2,4) U (infinity,4)

c) The x-coordinates of all points of inflection on the graph of f are the points where the concavity of the graph changes. We can see that a point of inflection occurs at x = -1 and x = 2