Q. 95 - Calculus

Question

Prove that if $f\text{'}\text{'}$ is zero on an interval, then f is linear on that interval.

Step-by-Step Solution

Answer

$f\left(x\right) = ax+b.$Hence, proved.

Step 1. Given Information.

Second derivative of the function is zero on an interval.

Step 2. Theorem

The Derivative Measures Where a Function is Increasing or Decreasing

Let f be a function that is differentiable on an interval I.

(a) If $f\text{'}$ is positive in the interior of I, then f is increasing on I.

(b) If $f\text{'}$ is negative in the interior of I, then f is decreasing on I.

(c) If $f\text{'}$ is zero in the interior of I, then f is constant on I.

Step 3. Proof

$$\begin{gathered} f\text{'}\text{'}\left(x\right) = 0, \\ Integrating w.r.t x we get, \\ f\text{'}\left(x\right) = a, where a is a cons\tan t. \\ Integrating w.r.t x once again we get, \\ f\left(x\right) = ax+b, where a and b both are cons\tan ts. \end{gathered}$$

Since $f\text{'}\text{'}$is zero it is sure that $f\text{'}$will be a constant and if first derivative is a constant then the function will be a linear function. Hence, Proved that the function will be linear.