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).

Answer 1

Given:  

show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3.),(2x^2, +,5, if x,>3.):}`

We have to show that the given function is differentiable at x = 3.

We have,

(LHD at x=3) = 

\[\lim_{x \to 3^-} \frac{f(x) - f(3)}{x - 3}\]

\[= \lim_{x \to 3} \frac{12x - 13 - 23}{x - 3}\]
\[ = \lim_{x \to 3} \frac{12x - 36}{x - 3}\]
\[ = \lim_{x \to 3} \frac{12 (x - 3)}{x - 3}\]
\[ = \lim_{x \to 3} 12 \]
\[ = 12\]

(RHD at x = 3) = 

\[\lim_{x \to 3^+} \frac{f(x) - f(3)}{x - 3}\]

\[= \lim_{x \to 3} \frac{2 x^2 + 5 - 23}{x - 3}\]
\[ = \lim_{x \to 3} \frac{2 x^2 - 18}{x - 3}\]
\[ = \lim_{x \to 3} \frac{2 ( x^2 - 9)}{x - 3}\]
\[ = \lim_{x \to 3} 2(x + 3) \]
\[ = 2 \times 6 \]
\[ = 12\]

Thus, (LHD at x=3) = (RHD at x=3) = 12.
So, 

\[f(x)\] is differentiable at x=3 and 

\[f'(3) = 12 .\]