Question

How do you know a function is decreasing or increasing at `x=1` given the function `4x^2-9x`?

Answer 1

Usually what people mean when they ask this question is "Is the slope of the tangent line at `x=1` positive of negative"? Or "Is the rate of change at `x=1` positive or negative"?

For `f(x)=4x^2-9x`, the derivative is `f'(x)=8x-9`

Whether we think of the derivative as the slope of the tangent line or the rate of change, it is clear that when `x=1`, the derivative is negative. (`f'(1)=8(1)-9=-1`)

This is generally explained by saying that,
at `x=1`, the function is decreasing at a rate of 1 (`f` unit) / (`x` unit).

(There is a bit of a conflict in terminology here. A function is constant at a single value of `x`. It is neither increasing nor decreasing. But in an interval containing a single value of `x`, the terms "increasing" and "decreasing" do apply.)