Question

If f and g are differentiable and \(f(x) \ge g(x)\)for \(a < x < b\),then \(f'(x) \ge g'(x)\) for \(a < x < b\).

Short answer

The answer is FALSE.

Step-by-step solution

Step 1: The Counterexample  

\(\begin{array}{l}f(x) = 0 \to f'(x) = 0\\g(x) = - {x^2} \to g'(x) = - 2x\\a = - 2\\b = 0\end{array}\)

The Illustration of the above assumption

By looking at the range (-2,0), it can be seen that \(f(x) \ge g(x)\) for \( - 2 < x < 0\), but \(f(x) < g(x)\) for \( - 2 < x < 0\).

Hence this disapproves the statement.