JEE Mathematics 2023 Question with Solution

Given: $f(x)=\max\{1+x+[x],\,2+x,\,x+2[x]\}$ for $0\le x\le 2$.

Find: The value of $(m+n)^2+2$, where $m$ is the number of discontinuity points in $[0,2]$ and $n$ is the number of non-differentiable points in $(0,2)$.

The greatest integer function $[x]$ changes value at integers, so split the interval at $x=1$ and check the endpoint values separately.

For $0\le x<1$, $[x]=0$. Then

$$f(x)=\max\{1+x,\,2+x,\,x\}=2+x$$

For $1\le x<2$, $[x]=1$. Then

$$f(x)=\max\{2+x,\,2+x,\,x+2\}=2+x$$

Thus, on both open intervals,

$$f(x)=2+x$$

Now check continuity at the junction points.

At $x=0$,

$$\lim_{x\to 0^+} f(x)=2,\quad f(0)=\max\{1,2,0\}=2$$

So $f$ is continuous at $x=0$.

At $x=1$,

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

So $f$ is continuous at $x=1$.

At $x=2$,

$$\lim_{x\to 2^-} f(x)=4,\quad f(2)=\max\{1+2+2,\,4,\,2+4\}=6$$

Since the left limit is not equal to the function value, $f$ is not continuous at $x=2$.

Hence,

$$m=1$$

For differentiability in $(0,2)$, the reduced form is $f(x)=2+x$ on both sides of $x=1$, so the derivative is the same on each side. Therefore $f$ is differentiable everywhere in $(0,2)$.

Hence,

$$n=0$$

Now compute

$$(m+n)^2+2=(1+0)^2+2=3$$

Therefore, the correct option is B.

Given: The function contains both $\max$ and $[x]$.

Find: The value of $(m+n)^2+2$.

The quickest approach is to first note that $[x]$ is constant on $[0,1)$ and $[1,2)$.

• On $[0,1)$, $[x]=0$, so the three expressions become $1+x$, $2+x$, and $x$. Their maximum is $2+x$.
• On $[1,2)$, $[x]=1$, so the three expressions become $2+x$, $2+x$, and $x+2$. Again the maximum is $2+x$.

So inside $(0,2)$, the function is just

$$f(x)=2+x$$

This immediately shows there is no non-differentiable point in $(0,2)$, so $n=0$.

Only endpoints and the jump point of $[x]$ need checking for continuity: $x=0,1,2$. At $x=0$ and $x=1$, the function matches the limit. At $x=2$,

$$\lim_{x\to 2^-} f(x)=4,\quad f(2)=6$$

So only one discontinuity occurs, giving $m=1$.

Therefore,

$$(m+n)^2+2=(1+0)^2+2=3$$

So the correct option is B.](streamdown:incomplete-link)