Q1246: Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0...

Class CBSE Class 12 Mathematics Continuity and Differentiability Q #1246

KNOWLEDGE BASED

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2 Marks 2024 AISSCE(Board Exam) VSA

Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}$ at $x=1.$

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AI Tutor Explanation

Detailed Solution

Step 1: Check continuity at x=1

For $f(x)$ to be differentiable at $x=1$, it must first be continuous at $x=1$. We need to check if $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$.

Step 2: Evaluate the left-hand limit

$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = (1)^2 + 1 = 2$

Step 3: Evaluate the right-hand limit

$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3-x) = 3 - 1 = 2$

Step 4: Evaluate the function at x=1

$f(1) = (1)^2 + 1 = 2$

Step 5: Check if the function is continuous

Since $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 2$, the function is continuous at $x=1$.

Step 6: Check differentiability at x=1

Now, we need to check if the left-hand derivative (LHD) is equal to the right-hand derivative (RHD) at $x=1$.

Step 7: Evaluate the left-hand derivative

LHD = $\lim_{h \to 0^-} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0^-} \frac{((1+h)^2 + 1) - 2}{h} = \lim_{h \to 0^-} \frac{1 + 2h + h^2 + 1 - 2}{h} = \lim_{h \to 0^-} \frac{2h + h^2}{h} = \lim_{h \to 0^-} (2 + h) = 2$

Step 8: Evaluate the right-hand derivative

RHD = $\lim_{h \to 0^+} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0^+} \frac{(3 - (1+h)) - 2}{h} = \lim_{h \to 0^+} \frac{3 - 1 - h - 2}{h} = \lim_{h \to 0^+} \frac{-h}{h} = -1$

Step 9: Check if the function is differentiable

Since LHD = $2$ and RHD = $-1$, LHD $\neq$ RHD. Therefore, the function is not differentiable at $x=1$.

Final Answer: Not differentiable at x=1

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the definition of differentiability and applying it to a piecewise function.

Knowledge Dimension: CONCEPTUAL

Justification: The question tests the understanding of the concept of differentiability, specifically how it relates to continuity and the existence of left-hand and right-hand derivatives.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's understanding of continuity and differentiability of functions, a core topic in calculus.