Question

Let $f\left(x\right)=x^3-x^2+x+1$
$g\left(x\right)=\{\begin{array}{c}max\{f\left(t\right),0\le t\le x\},0\le x\le 1\\ 3-x,1<x\le 2\end{array}$
Discuss the continuity and differentiability of the function $g\left(x\right)$ in the interval $(0,2)$.

• A. continuous everywhere
• B. differentiable everywhere
• C. continuous everywhere except $x=1$
• D. differentiable everywhere except $x=1$

Answer 1

Answer: (A), (D)

continuous everywhere
differentiable everywhere except $x=1$

$f\left(x\right)=x^3-x^2+x+1$
$f^{\prime }\left(x\right)=3x^2-2x+1$
We see that f'(x) > 0. $\therefore$ f(x) is strictly increasing.
Hence $g\left(x\right)=f\left(x\right)for0<x\le 1=3-xfor1<x\le 2$
Now we see that LHL = RHL at x=1.
But on differentiating LHL and RHL are not equal at x=1
Therefore options A,D are correct.