Question

Consider the function below, which has a relative minimum located at (-3 , -18) and a relative maximum located at (1/3, 14/27)

Answer 1

Since relative minimum is located at point (-3 , -18) and relative maximum is located at point (1/3, 14/27), then the function is:

1. strictly decreasing for $x\in (-\infty, -3)\cup ( \frac{1}{3}, \infty )$ and decreasing for $x\in (-\infty, -3]\cup [\frac{1}{3}, \infty )$

2. strictly increasing for $x\in (-3, \frac{1}{3} )$ and increasing for $x\in [-3, \frac{1}{3} ]$ .

Hence all points with $x\in (-\infty, -3]\cup [\frac{1}{3}, \infty )$ are located where the graph of f(x) is  decreasing. There are points (2, -18), (1, -2), (-3,-18) and (-4, -12). Check if they sutisfy the function expression:

1. $f(2)= -2^3 - 4\cdot 2^2 + 3\cdot 2=-8-16+6=-18$;

2. $f(1)= -1^3 - 4\cdot 1^2 + 3\cdot 1=-1-4+3=-2$;

3. $f(-4)= -(-4)^3 - 4\cdot (-4)^2 + 3\cdot (-4)=64-64-12=-12$.
Note that point (-3, -18) is a turning point.

Answer: ordered pairs (2, -18), (1, -2) and (-4, -12) are located where the graph of f(x) is strictly decreasing and (-3,-18) is located where the graph of f(x) is decreasing.