Let $f: R \rightarrow R$ be a function defined by $f(x) = \text{Min}\{x+1, |x|+1\}$. Then which of the following is true?

(C) $f(x)$ is differentiable everywhere

(A) $f(x) \geq 1$ for all $x \in R$

(B) $f(x)$ is not differentiable at $x = 1$

(D) $f(x)$ is not differentiable at $x = 0$

JEE Main · 2007 · 29 Apr 2007 · Offline · Continuity and Differentiability

Let

f : R \to R

be a function defined by

f (x) = Min {x + 1, ∣ x ∣ + 1}

. Then which of the following is true?

Answer: (C)

Compare the two expressions inside the minimum. For

x \geq 0

both

x + 1

and

∣ x ∣ + 1

equal

x + 1

. For

x < 0

∣ x ∣ + 1 = 1 - x

is larger than

x + 1

, so the smaller is again

x + 1

. Hence

f (x) = x + 1

for every real

x

, a straight line that is differentiable everywhere with slope

1

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option