CBSE Class 12 Mathematics Continuity and Differentiability Case Studies

Read the following and answer any four questions from (i) to (v).
A potter made a mud vessel, where the shape of the pot is based on \( f(x) = |x - 3| + |x - 2| \), where \( f(x) \) represents the height of the pot.

Question. When \( x > 4 \) What will be the height in terms of \( x \) ?
(a) \( x - 2 \)
(b) \( x - 3 \)
(c) \( 2x - 5 \)
(d) \( 5 - 2x \)
Answer: (c)
We have, \( f(x) = |x - 3| + |x - 2| \). When \( x > 4 \), \( f(x) = (x - 3) + (x - 2) = 2x - 5 \).

Question. Will the slope vary with \( x \) value?
(a) Yes
(b) No
(c) may or may not vary
(d) none of these
Answer: (a)
Yes, because when \( 2 < x < 3 \), we have \( f(x) = -(x - 3) + (x - 2) = 1 \). \( \Rightarrow \) Slope \( = f'(x) = 0 \). But when \( x > 3 \), we have \( f(x) = x - 3 + x - 2 = 2x - 5 \), then slope \( = f'(x) = 2 \).

Question. What is \( \frac{dy}{dx} \) at \( x = 3 \) ?
(a) 2
(b) -2
(c) Function is not differentiable
(d) 1
Answer: (c)
At \( x = 3 \):
\( \text{L.H.D} = \lim_{\lambda \to 0} \frac{f(3 - \lambda) - f(3)}{-\lambda} = \lim_{\lambda \to 0} \frac{-(3 - \lambda - 3) + (3 - \lambda - 2) - 1}{-\lambda} = \lim_{\lambda \to 0} \frac{\lambda + 1 - \lambda - 1}{-\lambda} = \lim_{\lambda \to 0} \frac{0}{-\lambda} = 0 \)
\( \text{R.H.D} = \lim_{\lambda \to 0} \frac{f(3 + \lambda) - f(3)}{\lambda} = \lim_{\lambda \to 0} \frac{(3 + \lambda - 3) + (3 + \lambda - 2) - 1}{\lambda} = \lim_{\lambda \to 0} \frac{\lambda + 1 + \lambda - 1}{\lambda} = 2 \)
L.H.D \( \neq \) R.H.D at \( x = 3 \). \( \therefore f(x) \) is not differentiable at \( x = 3 \).

Question. When the \( x \) value lies between (2, 3) then the function is
(a) \( 2x - 5 \)
(b) \( 5 - 2x \)
(c) 1
(d) 5
Answer: (c)
When \( 2 < x < 3 \), we have \( f(x) = -(x - 3) + (x - 2) = 1 \).

Question. If the potter is trying to make a pot using the function \( f(x) = [x] \), will he get a pot or not? Why?
(a) Yes, because it is a continuous function
(b) Yes, because it is not continuous
(c) No , because it is a continuous function
(d) No , because it is not continuous
Answer: (d)
We have the function \( f(x) = [x] \leq x \), where \( x \) is an integer. It is not a continuous function, so the potter can not make a pot using the function \( f(x) = [x] \).