Let , where denotes greatest integer function. If and are the number of points, where f is not continuous and is not differentiable, respectively, then equals __________. [2025]

Question

Let $f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ \min {1 + x + [x], x + 2 [x]}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix}$ , where $[\cdot]$ denotes greatest integer function. If $α$ and $β$ are the number of points, where f is not continuous and is not differentiable, respectively, then $α + β$ equals __________. [2025]

Smart Achievers · Accepted Answer

(5)

$f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ \min {1 + x + [x], x + 2 [x]}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix}$

$\Rightarrow$ $f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ x, & 0 \leq x < 1 \\ x + 2, & 1 \leq x < 2 \\ 5, & x > 2 \end{matrix}$

$∴$ By graph, we have f(x) is not continuous at $x \in {1, 2} \Rightarrow α = 2$

f(x) is not differentiable at $x \in {0, 1, 2} \Rightarrow β = 3$

$∴ α + β = 5$ .

Solution

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Solution

Q. Let f(x)={3x,x<0min {1+x+[x], x+2[x]},0≤x≤25,x>2, where [·] denotes greatest integer function. If α and β are the number of points, where f is not continuous and is not differentiable, respectively, then α+β equals __________. [2025]

Ans. (5) f(x)={3x,x<0min {1+x+[x], x+2[x]},0≤x≤25,x>2 ⇒ f(x)={3x,x<0x,0≤x<1x+2,1≤x<25,x>2 ∴ By graph, we have f(x) is not continuous at x∈{1,2} ⇒ α=2 f(x) is not differentiable at x∈{0,1,2} ⇒ β=3 ∴ α+β=5.

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