The function f(x) = |x| + 3 is

Correct Answer is : continuous on ℝ and differentiable on ℝ\{0}

continuous as well as differentiable on ℝ

continuous but not differentiable anywhere on ℝ

continuous on ℝ and differentiable on ℝ\{0}

Correct Answer is : continuous on ℝ and differentiable on ℝ\{0}

Test Index

UPSC NDA Math Model Paper 7

Question : 88 of 120

Marks: +1, -0

The function

f(x) = |x| + 3

Solution:

Given:

f(x) = |x| + 3

The given function can be re-written as:

f(x) = \begin{cases} 3 - x, & x < 0 \\ x + 3, & x \ge 0 \end{cases}

Let's examine the continuity of f(x) at x = 0

\Rightarrow \lim\limits_{x \rightarrow 0^{-}} f(x) = \lim\limits_{x \rightarrow 0} [3-(0-h)] = 3

\Rightarrow \lim\limits_{x \rightarrow 0^{+}} f(x) = \lim\limits_{x \rightarrow 0} [(h+0)+3] = 3

\Rightarrow \lim\limits_{x \rightarrow 0} f(x) = 3 = f(0) = 3

So, the given function f(x) is continuous on R.

Similarly, we can see that f(x) is differentiable over R\{0}

Hence, option D is the correct answer.

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