Question

must a function that is decreasing over a given interval always be negative over that same interval? explain.

Explanation:

Step1: Recall the definitions

A function \(y = f(x)\) is decreasing on an interval \((a,b)\) if for any \(x_1,x_2\in(a,b)\) with \(x_1

Step2: Provide a counter - example

Consider the function \(y=-x + 1\) on the interval \((0,2)\).
The derivative of \(y=-x + 1\) is \(y^\prime=-1<0\), so the function is decreasing on \((0,2)\).
When \(x = 0.5\), \(y=-0.5 + 1=0.5>0\).

Answer:

No. A decreasing function on an interval is not always negative on that interval. For example, the function \(y=-x + 1\) is decreasing on the interval \((0,2)\) (since its slope \(m=-1<0\)), but when \(x = 0.5\), \(y=-0.5 + 1=0.5>0\). So a decreasing function on an interval is not necessarily negative on that interval.