Question

Is `f(x)=(x-1)(x-1)(x-3)` increasing or decreasing at `x=1`?

Answer 1

Neither.

Explanation:

Rewrite `f(x)` as `x^3-5x^2+7x-3`.

Using the rule that `d/dx[x^n]=nx^(n-1)`.\

Therefore, `f'(x)=3x^2-10x+7`.

The derivative gives us the slope of the function at a point. If the derivative is positive, then the slope is positive, so the function is increasing. If the derivative is negative, the slope is negative, so the function is decreasing. Plug in `x=1` to get

`f'(1)=3(1^2)-10x(1)+7=0`

So the slope at `x=1` is `0`, so the function is neither increasing nor decreasing at `x=1`. graph{(x-1)^2(x-3) [-6.24, 6.247, -3.12, 3.12]}