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Question
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Solution
Given:
We have to show that the given function is differentiable at x = 3.
We have,
(LHD at x=3) =
\[= \lim_{x \to 3} \frac{12x - 13 - 23}{x - 3}\]
\[ = \lim_{x \to 3} \frac{12x - 36}{x - 3}\]
\[ = \lim_{x \to 3} \frac{12 (x - 3)}{x - 3}\]
\[ = \lim_{x \to 3} 12 \]
\[ = 12\]
(RHD at x = 3) =
\[= \lim_{x \to 3} \frac{2 x^2 + 5 - 23}{x - 3}\]
\[ = \lim_{x \to 3} \frac{2 x^2 - 18}{x - 3}\]
\[ = \lim_{x \to 3} \frac{2 ( x^2 - 9)}{x - 3}\]
\[ = \lim_{x \to 3} 2(x + 3) \]
\[ = 2 \times 6 \]
\[ = 12\]
Thus, (LHD at x=3) = (RHD at x=3) = 12.
So,
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