Question

For what values of x is `f(x)=(x-2)(x-7)(x-3)` concave or convex?

Answer 1

Concave on `(-oo,4)`; convex on `(4,+oo)`

Explanation:

The concavity and convexity of a function are determined by the sign (positive/negative) of the second derivative.

• If `f''(a)<0`, then `f(x)` is concave at `x=a`.
• If `f''(a)>0`, then `f(x)` is convex at `x=a`.

In order to find the second derivative, we should first simplify the undifferentiated function by distributing.

`f(x)=(x^2-9x+14)(x-3)=x^3-12x^2+41x-42`

Now, find the first and second derivatives through a simple application of the power rule.

`f'(x)=3x^2-24x+41`
`f''(x)=6x-24`

Now, we must find the times when `6x-24` is positive and when it is negative. The times when the function could shift from positive to negative or vice versa are when `6x-24=0`.

`6x-24=0`
`6x=24`
`x=4`

The sign of the second derivative, and by extension, the concavity/convexity of the function, could shift only at `x=4`. Thus, we should test points on either side of `x=4` to determine which concavity/convexity is present on either side.

When `mathbf(x <4)`:

Test point at `x=0`:

`f''(0)=6(0)-24=-24`

Since this is `<0`, the entire interval `(-oo,4)` will be concave.

When `mathbf(x >4)`:

Test point at `x=5`:

`f''(5)=6(5)-24=6`

Since this is `>0`, the entire interval `(4,+oo)` will be convex.

We can check the graph of `f(x)`: convexity is characterized by a `uu` shape, and concavity is characterized by a `nn` shape. The concavity of the graph should shift at `x=4`.

graph{x^3-12x^2+41x-42 [-2, 10, -30, 15]}