Mathematics - Continuity - Differentiability Question with Solution | TestHub

Take $p=(x,\varnothing (x));Q=(y,\varnothing (y))$ be any two points the curve $y=\varnothing (x)$

Let ' R ' divides the line segment $\overline{PQ}$ in the ratio 2:1 then $R=\left(\frac{x+2y}{3},\frac{\varnothing (x)+2\varnothing (y)}{3}\right)$

Clearly TM $>$ RM

$\Rightarrow \varnothing \left(\frac{x+2y}{3}\right)>\frac{\varnothing (x)+2\varnothing (y)}{3}$

Equality holds iff $\varnothing (x)$ is a linear function.

$$\begin{array}{lll}\therefore \varnothing (x)=ax+b& & \\ & & \end{array}$$

$$\because {\varnothing }^{\prime }(0)=1\Rightarrow a=1$$

$$\because \varnothing (0)=2$$

$$\because \varnothing (x)=x+2$$