Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by $f (x) = [x^{2} - 3]$ and $g (x) = | x | f (x) + | 4 x - 7 | f (x)$ , where $[y]$ denotes the greatest integer less than or equal to y for $y \in R$ . Then

f is discontinuous exactly at three points in

[- \frac{1}{2}, 2]

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f is discontinuous exactly at four points in

[- \frac{1}{2}, 2]

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g is NOT differentiable exactly at four points in

[- \frac{1}{2}, 2]

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g is NOT differentiable exactly at five points in

[- \frac{1}{2}, 2]

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Solution

The correct options are
B f is discontinuous exactly at four points in $[- \frac{1}{2}, 2]$
C g is NOT differentiable exactly at four points in $[- \frac{1}{2}, 2]$
$f (x) = [x^{2} - 3] = [x^{2}] - 3 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - 3 & - \frac{1}{2} \leq x < 1 - 2 & 1 \leq x < \sqrt{2} - 1 & \sqrt{2} \leq x < \sqrt{3} 0 & \sqrt{3} \leq x < 2 1 & x = 2 \end{matrix}$
$g (x) = | x | f (x) + | 4 x - 7 | f (x)$
$(| x | + | 4 x - 7 |) [x^{2} - 3] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} (- x - 4 x - 7) (- 3) & - \frac{1}{2} \leq x < 0 (x - (4 x - 7)) (- 3) & 0 \leq x < 1 (x - (4 x - 7)) (- 2) & 1 \leq x < \sqrt{2} (x - (4 x - 7)) (- 1) & \sqrt{2} \leq x < \sqrt{3} ((x - (4 x - 7)) (0) & \sqrt{3} \leq x < 7 / 4 (x + (4 x - 7)) (0) & 7 / 4 \leq x < 2 (x + (4 x - 7)) (1) & x = 2 \end{matrix}$
$= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 15 x + 21 & - \frac{1}{2} 9 x - 21 & 0 \leq x < 1 6 x - 14 & 1 \leq x < \sqrt{2} 3 x - 7 & \sqrt{2} \leq x < \sqrt{3} 0 & \sqrt{3} \leq x < 2 5 x - 7 & x = 2 \end{matrix}$
Now graph of given function is clearly $F$ is not discontinuous at exactly $4$ point in $[\frac{- 1}{2}, 2]$ and $g$ is not differentiable at $4$ points in $(\frac{- 1}{2}, 2)$
Hence Ans. are $B C$

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Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by
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L e t f : [- \frac{1}{3}, 3] \to R a n d g : [- \frac{1}{3}, 3] \to R d e f i n e d b y f (x) = [x^{2} - 4] a n d g (x) = | x - 2 | f (x) + | 3 x - 5 | f (x)

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Let f:[−12,2]→R and g:[−12,2]→R be functions defined by f(x)=[x2−3] and g(x)=|x|f(x)+|4x−7|f(x), where [y] denotes the greatest integer less than or equal to y for y∈R. Then

Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by $f (x) = [x^{2} - 3]$ and $g (x) = | x | f (x) + | 4 x - 7 | f (x)$ , where $[y]$ denotes the greatest integer less than or equal to y for $y \in R$ . Then