Question

For what values of x is `f(x)= -x^3+3x^2-2x+2` concave or convex?

Answer 1

Concave (Convex) Up on the interval `( -oo,1 )`
Concave (Convex) Down on the interval `( 1, oo )`

Explanation:

We are given the function `f(x) = -x^3 + 3x^2 - 2x + 2`

`color(red)(Step.1)`

Find the First Derivative

`f'(x) = -3x^2 + 6x -2`

`color(red)(Step.2)`

Find the Second Derivative

`f''(x) = -6x + 6`

`color(red)(Step.3)`

Next, set

`f''(x) = -6x + 6 = 0`

Simplifying, we get `x = 1`

`color(red)(Step.4)`

Then, we consider a number larger than 1 and a number smaller than 1 and substitute the values in our Second Derivative.

If the number is Greater than 1, our `f''(x) = -6x + 6"` will yield a "Negative" number.

If the number is Less than than 1, our `f''(x) = -6x + 6"` will yield a "Positive" number.

Hence, we observe that `f(x)` is "Concave Up" on the interval `(-oo, 1)` and "Concave Down" on the interval `(1, oo)`

Refer to the Number Line as shown below:

                                                    1

:..........................................................*..................................................................:

             Positive                                                  Negative