Question

The function defined by f(x) = {(|x - 3|; x ≥ 1), (1/4x² - 3/2x  + 13/4; x < 1) is

(A) Continuous at x = 1

(B) Continuous at x = 3

(C) Differentiable at x = 1

(D) All the above 

Answer 1

Answer is (D) All the above

Since |x - 3| = x - 3, if x ≥ 3;

|x - 3| = -x + 3, if x < 3

Hence, the given function can be defined as

 

Now proceed to check the continuity and differentiability at x = 1.

lim_(x →1) f(x) = f(1) = 2

and lim_(x →1) f(x) = - 1

So, f(x) is continuous and differentiable at x = 1.

Also,

lim_(x →3) = f(3) = 0

So, f(x) is also continuous at x = 3.