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

1 Answer
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(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.)