Q. 87 - Calculus

Question

Prove that every nonconstant linear function is either always increasing or always decreasing.

Step-by-Step Solution

Answer

It is proved that every nonconstant linear function is either always increasing or always decreasing.

Step 1. Given Information

We are given that every nonconstant linear function is either always increasing or always decreasing.

Step 2. Proving the statement

Consider a function of the form,

$f\left(x\right)=mx+b\text{; where }m\ne 0\text{. }$

That is f is a non-constant linear function.

The first derivative of the function is,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}(mx+b) \\ =m·1+0 \\ =m \end{gathered}$$

Therefore, f is either positive or negative.

If m is positive, the function is increasing. If m is negative then the function is decreasing. So, every non constant linear function is either always increasing or always decreasing.