Question

Is `f(x)=(x-2)(x+5)(x+2)` increasing or decreasing at `x=-3`?

Answer 1

`f(x)=(x-2)(x+5)(x+2)` is decreasing at `x=-3`

Explanation:

We are given `f(x)=(x-2)(x+5)(x+2)`

Now the function is increasing at a given point, if `(df)/(dx)>0` at that point and is decreasing if `(df)/(dx)<0`.

Here `f(x)=(x-2)(x+5)(x+2)=(x+5)(x^2-4)=x^3+5x^2-4x-20`

and `(df)/(dx)=3x^2+10x-4`

at `x=-3`, we have `((df)/(dx))_(x=-3)=3(-3)^2+10(-3)-4`

= `27-30-4=-7`

Hence `f(x)=(x-2)(x+5)(x+2)` is decreasing at `x=-3`

(see graph not drawn to scale - shrunk alon `y`-axis)
graph{(x-2)(x+5)(x+2) [-10, 10, -40, 40]}