Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

12. If\(f\)and\(g\)are increasing on an interval\(I\), then\(f - g\)is increasing on\(I\).

Short answer

The given statement is false.

Step-by-step solution

Increasing Function

A function is said to be an increasing function on an interval I if for any two numbers x and y such that \(x < y\) implies \(f\left( x \right) \le f\left( y \right)\).

Example for false statement

Let \(f\left( x \right) = x\) and \(g\left( x \right) = {x^2}\) be an increasing functions on the interval (1,2).

The derivative of the function f and g is \(f'\left( x \right) = 1\) and \(g'\left( x \right) = 2x\). Then the required function f-g is \(h\left( x \right) = 1 - 2x\).

Since, \(h\left( x \right) = - 1\) at \(x = 1\) and \(h\left( x \right) = - 3\) at \(x = 2\) implies that the function is decreasing on the interval (1,2).

Therefore, the given statement is false.