Let m and n be the number of points at which the function f(x)= $\max\{x, x^3, x^5, \dots, x^{21}\}, x\in\mathbb{R}$ is not differentiable and not continuous, respectively. Then m + n is equal to _____

Step 1: Analyze the function for continuity

The function $f(x)$ is defined as the maximum of a finite set of continuous functions. The maximum of a finite number of continuous functions is always continuous. Therefore, $f(x)$ is continuous for all $x \in \mathbb{R}$. This means the number of points where $f(x)$ is not continuous, denoted by $n$, is $0$.

Step 2: Identify the dominant term for different intervals of x

To find the points of non-differentiability, we need to determine which term in the set $\{x, x^3, \dots, x^{21}\}$ is the maximum for different intervals of $x$. We compare the magnitudes of $x^k$ for different odd powers $k$. For $x \in (0, 1)$, $x > x^3 > \dots > x^{21}$. For $x \in (-1, 0)$, $x < x^3 < \dots < x^{21}$. For $x > 1$, $x^{21} > x^{19} > \dots > x$. For $x < -1$, $x^{21} < x^{19} < \dots < x$. At $x=1$ and $x=-1$, all terms are equal.

Step 3: Define f(x) piecewise

Based on the comparison of terms, we can write $f(x)$ as a piecewise function. When $x \le -1$, $x$ is the largest. When $-1 < x < 0$, $x^{21}$ is the largest. When $0 \le x < 1$, $x$ is the largest. When $x \ge 1$, $x^{21}$ is the largest. Note that at $x=0$, $x=0$ and $x^{21}=0$, so $f(0)=0$.

Step 4: Check differentiability at critical points

A function defined as the maximum of continuous functions is not differentiable at points where the 'active' function changes, provided the derivatives from the left and right are different. We check the differentiability at $x = -1, 0, 1$. At $x=-1$, the left derivative is $1$ (from $x$) and the right derivative is $21$ (from $x^{21}$), so it's not differentiable. At $x=0$, the left derivative is $0$ (from $x^{21}$) and the right derivative is $1$ (from $x$), so it's not differentiable. At $x=1$, the left derivative is $1$ (from $x$) and the right derivative is $21$ (from $x^{21}$), so it's not differentiable.

Step 5: Calculate m + n

We found that $f(x)$ is not differentiable at $x = -1, 0, 1$. Thus, the number of points where $f(x)$ is not differentiable, $m$, is $3$. We also found that $f(x)$ is continuous everywhere, so the number of points where $f(x)$ is not continuous, $n$, is $0$. Therefore, $m + n = 3 + 0 = 3$.

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