​ The function f(x) = (1+x, when x ≤ 2) (5 - x, when x > 2) is ​A. continuous as well as differentiable at x = 2

Question

The function f(x) = \(\begin{cases} 1+x,when\;x\leq2 \\ 5-x,when\;x>2 \end{cases}\) is

f(x) = (1+x, when x ≤ 2), (5 - x, when x > 2)

A. continuous as well as differentiable at x = 2

B. continuous but not differentiable at x = 2

C. differentiable but not continuous at x = 2

D. none of these

Answer 1

Answer is : B. continuous but not differentiable at x = 2

For continuity left hand limit must be equal to right hand limit and value at the point.

Continuity at x = 2.

For continuity at x = 2,

f(2) = 1+2 = 3

∴ f(x) is continuous at x = 2

Now for differentiability.

= \(\lim\limits_{h\rightarrow0}\frac{h}{-h}\)

= -1

As, f’(2^-) is not equal to f(2⁺)

∴ f(x) is not differentiable.