# For what values of x is #f(x)= -9x^3 + 4 x^2 + 7x -2 # concave or convex?

##### 2 Answers

#### Explanation:

A function is **concave** (or concave down) where its *derivative* is decreasing. Graphically, a concave region looks like a cave (or cut from a cave shape). (Quick example: the function

A function is **convex** (or concave up) where its *derivative* is increasing. Graphically, a convex region looks like a V (or cut from a V shape). (The function

To find these regions, we need to analyze the behaviour of the function's derivative. Hence, we need to find the *derivative* of the derivative:

Given

#f'(x)="–"27x^2+8x+7#

by the power rule, and so the derivative of this is

#f''(x)="–"54x+8# .

Just like how we know

#f''(x)="–"54x+8 > 0#

#=>" –"54x>"–8"#

#=>" "x<8/54=4/27# .

So **inflection point**—a point at which the direction of concavity changes.

Finding where

Here's a graph of the function, with the inflection point circled.

graph{(-9x^3+4x^2+7x-2-y)((x-4/27)^2+(y+0.9)^2/36-0.0025)=0 [-2.1, 2.1, -6, 6]}

As we come from

Notice how the function is V-shaped to the left of the inflection point, and cave-shaped to its right. That's an easy way to remember the difference between concave and convex: concave is like a cave; convex is like a V.

#### Explanation:

We calculate the first derivative and build a chart of variations

To determine the critical points, we solve the equation

As,

The chart of variations is

So,

and concave when