Question

Let $f:[a,b]\to [1,\infty )$ be a continuous function and let $g:R\to R$ be defined as $g\left(x\right)=\{\begin{array}{ccc}0& if& x<a,\\ {\int }_a^{x}f\left(t\right)dt& if& a\le x\le b,\\ {\int }_a^{b}f\left(t\right)dt& if& x>b\end{array}$. Then

• A. $g\left(x\right)$ is continuous but not differentiable at $a$
• B. $g\left(x\right)$ is differentiable on $R$
• C. $g\left(x\right)$ is continuous but not differentiable at $b$
• D. $g\left(x\right)$ is continuous and differentiable at either $a$ or $b$ but not both

Answer 1

Answer: (A), (C)

$g\left(x\right)$ is continuous but not differentiable at $a$
$g\left(x\right)$ is continuous but not differentiable at $b$

Since $f\left(x\right)\ge 1\forall x\in [a,b]$
for g(x)
$LHDatx=a$ is zero
and $RHDat(x=a)={lim}_{x\to a^{+}}\frac{{\int }_a^{x}f\left(t\right)dt-0}{x-a}=\underset{x-a^{+}}{lim}f\left(x\right)\ge 1$
Hence, g(x) is not differentiable at x$=$ a
Similarly LHD at x $=$ b is greater than 1
g(x) is not differentiable at x $=$ b