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

Mar 22, 2015

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 $f \left(x\right) = 4 {x}^{2} - 9 x$, the derivative is $f ' \left(x\right) = 8 x - 9$

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. ($f ' \left(1\right) = 8 \left(1\right) - 9 = - 1$)

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.)