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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY1 Mark
Question
Choose the correct answers from the given four options:
The function $\text{f(x)}=\text{e}^{|\text{x}|}$ is:
  1. Continuous everywhere but not differentiable at x = 0.
  2. Continuous and differentiable everywhere.
  3. Not continuous at x = 0.
  4. None of these.
✓

Answer

  1. Continuous everywhere but not differentiable at x = 0.

Solution:

Let $\text{u(x)}=|\text{x}|$ and $\text{v(x)}=\text{e}^\text{x}$

$\therefore\ \text{f(x)}=\text{vou(x)}=\text{v}[\text{u(x)]}$

$=\text{v}|\text{x}|=\text{e}^{|\text{x}|}$

Since, u(x) and v(x) are both continuous functions.

So, f(x) is also continuous function but u(x) = |x| is not differentiable at x = 0, whereas v(x) = ex is differentiable at everywhere.

Hence, f(x) is continuous everywhere but not differentiable at x = 0.

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