JEE Advanced 2014 · Paper 1 · Mathematics Q45

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MathematicsMultiple correct+3Question 45 of 60

Let $f:[a,b]\to[1,\infty)$ be a continuous function and let $g:\mathbb R\to\mathbb R$ be defined as
$g(x)=\begin{cases}0&\text{if }x<a,\\\displaystyle\int_a^xf(t)\,dt&\text{if }a\le x\le b,\\\displaystyle\int_a^bf(t)\,dt&\text{if }x>b.\end{cases}$
Then

g(x)

is continuous but not differentiable at

a

g(x)

is differentiable on

\mathbb R

g(x)

is continuous but not differentiable at

b

g(x)

is continuous and differentiable at either

a

b

but not both

Answer and worked solution

Official answer: A, C

Continuity. $g$ is 0 to the left of $a$ , $\int_a^xf$ on $[a,b]$ (which is 0 at $a$ ), and constant $\int_a^bf$ beyond $b$ . The pieces join, so $g$ is continuous everywhere.

Derivatives at the joins.

At $a$ : left derivative 0, right derivative $f(a)\ge1$ . Not differentiable.
At $b$ : left derivative $f(b)\ge1$ , right derivative 0. Not differentiable.
(A) True.
(B) False.
(C) True.
(D) It is differentiable at neither. False.

(A), (C)

Topics:Differentiability Fundamental theorem of calculus

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JEE Advanced 2014 · Paper 1, Mathematics question 45