Question

1. the functions f(x) and g(x) are each defined for -3 ≤ x ≤ 4. the graph of f(x) is provided above, and a table is provided below describing the rate of change of g on select intervals.

interval of x	-3 < x < -1	-1 < x < 1	1 < x < 3	3 < x < 4
rate of change of g	positive and decreasing	negative and increasing	positive and increasing	positive and decreasing

(a) at which x - value does g(x) reach a local maximum? explain.
(b) the function f has a rate of change that is negative and decreasing on the interval a ≤ x ≤ b. identify the values of a and b.
(c) (i) find the intervals of x on which f(x) and g(x) have the same concavity.
(ii) justify your intervals.

Explanation:

Step1: Recall the condition for local maximum

A function $g(x)$ has a local maximum when its rate of change changes from positive to negative. From the table of the rate - of - change of $g(x)$, the rate of change of $g(x)$ is positive on $-3 < x<-1$ and negative on $-1 < x < 1$. So $g(x)$ reaches a local maximum at $x=-1$.

Step2: Analyze the rate of change of $f(x)$

The rate of change of a function is given by its derivative. The rate of change of $f(x)$ is negative and decreasing when the slope of the tangent line to the graph of $y = f(x)$ is negative and becoming more negative. Looking at the graph of $f(x)$, this occurs on the interval $1\leq x\leq3$.

Step3: Recall the concept of concavity

The concavity of a function is related to the second - derivative. A function is concave up when its second - derivative is positive (the rate of change of the first - derivative is positive) and concave down when its second - derivative is negative (the rate of change of the first - derivative is negative).
For $f(x)$:

• The function $f(x)$ is concave up when the slope of the tangent line is increasing. From the graph, $f(x)$ is concave up on $(-3,-2)$ and $(1,2)$. It is concave down on $(-2,1)$ and $(2,4)$.
• For $g(x)$:
• On $-3 < x<-1$, the rate of change of $g(x)$ is positive and decreasing, so $g(x)$ is concave down.
• On $-1 < x < 1$, the rate of change of $g(x)$ is negative and increasing, so $g(x)$ is concave up.
• On $1 < x < 3$, the rate of change of $g(x)$ is positive and increasing, so $g(x)$ is concave up.
• On $3 < x < 4$, the rate of change of $g(x)$ is positive and decreasing, so $g(x)$ is concave down.
• The intervals where $f(x)$ and $g(x)$ have the same concavity are $-1 < x < 1$ and $3 < x < 4$.

Answer:

(A) $x = - 1$ because the rate of change of $g(x)$ changes from positive to negative at $x=-1$.
(B) $a = 1$, $b = 3$ since the slope of the tangent line to $y = f(x)$ is negative and becoming more negative on this interval.
(C) (i) $-1 < x < 1$ and $3 < x < 4$; (ii) Analyzed the concavity of $f(x)$ from its graph and the concavity of $g(x)$ from the rate - of - change information in the table.