Question

For each of the following functions, respond to the given prompts. Show all work that leads to your responses. 

Given: 𝑓(𝑥) = 3𝑥^3 − 18𝑥^2 − 45𝑥 + 10 

a. On what interval(s), is 𝑓(𝑥) increasing? Show all work and explain

b. At what value(s) of 𝑥 does 𝑓(𝑥) have a relative minimum? Show all work and explain

c) On what interval(s), is 𝑓(𝑥) decreasing and concave up? Show all work and explain

Answer 1

Given: $𝑓(𝑥) = 3𝑥^3 − 18𝑥^2 − 45𝑥 + 10$

a. On what interval(s), is 𝑓(𝑥) increasing?

$f'(x) > 0$

$9x^2-36x-45 = 9(x-4x-5) = 9(x-5)(x+1) > 0$

function is increasing where $(-\infin; -1)\cup (5; \infin)$

b. At what value(s) of 𝑥 does 𝑓(𝑥) have a relative minimum?

We say that f(x) has a relative minimum at x = c if $f(x) \geqslant f(c)$ for every x in some open interval around $x = c$ x = -1 is relative minimum

c) On what interval(s), is 𝑓(𝑥) decreasing and concave up?

$f'(x) < 0$

$9x^2-36x-45 = 9(x-4x-5) = 9(x-5)(x+1) < 0$

function is decreasing where $(-1;5)$

• When the second derivative is positive, the function is concave upward.

$f''(x) = 18x-36 > 0$

$x > 2$

function is concave up where $(2; \infin)$