Question

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

Short answer

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

Step-by-step solution

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

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=0, \\ Integratingw.r.txweget, \\ f\text{'}\left(x\right)=a,whereaisacons\tan t. \\ Integratingw.r.txonceagainweget, \\ f\left(x\right)=ax+b,whereaandbbotharecons\tan ts. \end{gathered}$$

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.