Find the open intervals where the function f(x) = 3x^3 + 9x^2 + 175x - 6 is concave upward or...
Question:
Find the open intervals where the function {eq}f(x) = 3x^3 + 9x^2 + 175x - 6 {/eq} is concave upward or concave downward. Find any inflection points.
Concavity:
We recall that if {eq}f''(x)>0 {/eq} on {eq}I {/eq}, then {eq}f(x) {/eq} is concave up on {eq}I {/eq}.
If {eq}f''(x)<0 {/eq} on {eq}I {/eq}, then {eq}f(x) {/eq} is concave down on {eq}I {/eq}.
An inflection point is any point in the domain of the curve where the concavity changes.
Answer and Explanation:
Let {eq}f(x)=3x^3+9x^2+175x-6 {/eq}. We then have
{eq}f'(x)=9x^2+18x+175\\ f''(x)=18x+18 {/eq}
We can then see that {eq}f''(x)=0 {/eq} when {eq}x=-1 {/eq}.
It follows that {eq}f''(x)<0 {/eq} on {eq}(-\infty,-1) {/eq} and {eq}f''(x)>0 {/eq} on {eq}(-1,\infty) {/eq}.
So we have {eq}f(x) {/eq} is concave up on {eq}(-1,\infty) {/eq} and concave down on {eq}(-\infty,-1) {/eq}. We also conclude that {eq}f(x) {/eq} has an inflection point at {eq}x=-1 {/eq}.
Learn more about this topic:
from
Chapter 9 / Lesson 5You might not think of a cup when you think of an awesome skateboard ramp. But I'm sure a really bad ramp would give you a frown, right? Learn about cups and frowns in this lesson on concavity and inflection points.
Related to this Question
