Key Concepts and Formulas

• Greatest Integer Function $[x]$: The greatest integer less than or equal to $x$. Key property: $[x]$ is constant on intervals of the form $[k, k+1)$ where $k$ is an integer.
• Maximum Function $\max\{a, b, c\}$: The largest among the given values. To determine the maximum, we compare the expressions for different intervals of $x$.
• Continuity: A function $f(x)$ is continuous at a point $c$ if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$. Points of discontinuity often occur where the definition of the function changes, especially when the greatest integer function is involved.
• Differentiability: A function $f(x)$ is differentiable at a point $c$ if the limit of the difference quotient exists: $\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$. A necessary condition for differentiability is continuity. Points where the function has sharp corners, cusps, or vertical tangents are points of non-differentiability. For functions involving the greatest integer function, points where $[x]$ changes value (i.e., integers) are candidates for non-differentiability.

Step-by-Step Solution

The problem asks us to find the number of points of discontinuity ($m$) in $[0,2]$ and the number of points of non-differentiability ($n$) in $(0,2)$ for the function $f(x)=\max \{1+x+[x], 2+x, x+2[x]\}$, for $0 \leq x \leq 2$.

Step 1: Analyze the behavior of $[x]$ in the given interval. The interval for $x$ is $[0, 2]$. The greatest integer function $[x]$ changes its value at integer points. In the interval $[0, 2]$, the integer points are $0, 1, 2$. We will consider the intervals based on the values of $[x]$:

• For $x \in [0, 1)$, $[x] = 0$.
• For $x \in [1, 2)$, $[x] = 1$.
• For $x = 2$, $[x] = 2$.

Step 2: Define the three functions involved in the maximum for each interval. Let the three functions be: $g(x) = 1+x+[x]$ $h(x) = 2+x$ $k(x) = x+2[x]$

We will evaluate these functions in the relevant intervals:

• For $x \in [0, 1)$: $[x] = 0$. $g(x) = 1+x+0 = 1+x$ $h(x) = 2+x$ $k(x) = x+2(0) = x$

• For $x \in [1, 2)$: $[x] = 1$. $g(x) = 1+x+1 = 2+x$ $h(x) = 2+x$ $k(x) = x+2(1) = x+2$

• For $x = 2$: $[x] = 2$. $g(x) = 1+2+2 = 5$ $h(x) = 2+2 = 4$ $k(x) = 2+2(2) = 2+4 = 6$

Step 3: Determine $f(x)$ by finding the maximum of the three functions in each interval.

• For $x \in [0, 1)$: We need to find $\max\{1+x, 2+x, x\}$. Comparing $1+x$ and $2+x$: Since $2+x > 1+x$ for all $x$. Comparing $2+x$ and $x$: Since $2+x > x$ for all $x$. Therefore, for $x \in [0, 1)$, $f(x) = \max\{1+x, 2+x, x\} = 2+x$.

• For $x \in [1, 2)$: We need to find $\max\{2+x, 2+x, x+2\}$. Note that $2+x$ and $x+2$ are the same function. So, for $x \in [1, 2)$, $f(x) = \max\{2+x, 2+x, x+2\} = 2+x$.

• For $x = 2$: We need to find $\max\{g(2), h(2), k(2)\}$. $g(2) = 5$ $h(2) = 4$ $k(2) = 6$ So, for $x = 2$, $f(2) = \max\{5, 4, 6\} = 6$.

Combining these results, we can define $f(x)$ piecewise: $f(x) = \begin{cases} 2+x & \text{if } 0 \leq x < 2 \\ 6 & \text{if } x = 2 \end{cases}$

Step 4: Determine the number of points of discontinuity ($m$) in $[0, 2]$. A function can be discontinuous at points where its definition changes. In our case, the definition changes at $x=2$. We need to check the continuity at $x=2$. For continuity at $x=2$, we require $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)$.

• Left-hand limit at $x=2$: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (2+x) = 2+2 = 4$.

• Value of the function at $x=2$: $f(2) = 6$.

Since $\lim_{x \to 2^-} f(x) \neq f(2)$ ($4 \neq 6$), the function $f(x)$ is discontinuous at $x=2$. For $x \in [0, 2)$, $f(x) = 2+x$, which is a linear function and thus continuous on this interval. Therefore, the only point of discontinuity in $[0, 2]$ is $x=2$. So, $m = 1$.

Step 5: Determine the number of points of non-differentiability ($n$) in $(0, 2)$. The function $f(x)$ is defined as $f(x) = 2+x$ for $0 \leq x < 2$. This is a linear function with a constant slope of 1. A linear function is differentiable everywhere. Therefore, $f(x) = 2+x$ is differentiable for all $x \in [0, 2)$. The derivative of $f(x) = 2+x$ is $f'(x) = 1$. This derivative exists for all $x$ in $(0, 2)$. At $x=2$, the function is discontinuous, so it cannot be differentiable there. We are looking for points of non-differentiability in the open interval $(0, 2)$. Since $f(x) = 2+x$ for all $x \in [0, 2)$, and this function is differentiable with a derivative of 1 everywhere in this interval, there are no points of non-differentiability in $(0, 2)$. So, $n = 0$.

Step 6: Calculate the final expression $(m+n)^2+2$. We found $m=1$ and $n=0$. $(m+n)^2+2 = (1+0)^2+2 = (1)^2+2 = 1+2 = 3$.

Common Mistakes & Tips

• Incorrectly evaluating the maximum: Carefully compare all the expressions for $f(x)$ in each interval. A small error here can propagate through the entire solution.
• Confusing continuity and differentiability: Remember that differentiability implies continuity, but continuity does not imply differentiability. Points of discontinuity are automatically points of non-differentiability.
• Intervals for $[x]$: Pay close attention to the endpoints of the intervals when dealing with the greatest integer function. For example, $[x]$ is $0$ for $x \in [0, 1)$, not $[0, 1]$.
• Differentiability at integer points: Integer points are critical for functions involving $[x]$ as they are potential points of non-differentiability due to sharp corners or discontinuities.

Summary

We first analyzed the greatest integer function $[x]$ in the given interval $[0,2]$ and defined the three component functions. By comparing these functions over the intervals $[0,1)$ and $[1,2)$, and at $x=2$, we determined the piecewise definition of $f(x)$. We then checked for continuity at $x=2$ by comparing the left-hand limit with the function value, finding one point of discontinuity ($m=1$). Next, we examined differentiability in the interval $(0,2)$. Since $f(x)=2+x$ on $[0,2)$, which is a differentiable linear function, there are no points of non-differentiability in $(0,2)$ ($n=0$). Finally, we computed $(m+n)^2+2 = (1+0)^2+2 = 3$.

The final answer is $\boxed{3}$.