(i) Find the R.H.D. of f(x) at x = 1

Question

Read the following passage and answer the questions given below. 

Let f(x) be a real valued function, then its

Left Hand Derivative (L.H.D.) :

\(Lf'(a)= \lim\limits_{h \to 0} \frac {f(a-h)-f(a)}{-h}\)

Right Hand Derivative (R.H.D.) :

\(Rf' (a) =\lim\limits_{h \to 0}\frac {f(a +h)-f(a)}{h}\)

Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal.

For the function f(x) = \(\begin{cases} |x-3| , & \quad x≥1\\ \frac{x^2}{4}-\frac {3x}{2} + \frac {13}{4}, & \quad x<1 \end{cases},\) answer the following questions. 

(i) Find the R.H.D. of f(x) at x = 1 

(ii) Find the L.H.D. of f(x) at x = 1 

(iii) Find the value of x where f(x) is non-differentiable.

OR 

(iii) Find the value of f′(2).

Answer 1

(iii) Since, R.H.D. at x = 3 is 1 and L.H.D. at x = 3 is –1 

∴ f(x) is non-differentiable at x = 3.

OR 

(iii) f (x) is continuous in the interval 1 ≤ x < 3

There is no break point in the interval 1 ≤ x < 3 

So, f(x) is continuous at x = 2