Let m and n be the number of points at which the function f(x)= max{x,x3,x5,…,x21},x∈R is not differentiable and not continuous, respectively. Then m + n is equal to _____
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Step-by-Step Solution
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∈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,x3,…,x21} is the maximum for different intervals of x. We compare the magnitudes of xk for different odd powers k. For x∈(0,1), x>x3>⋯>x21. For x∈(−1,0), x<x3<⋯<x21. For x>1, x21>x19>⋯>x. For x<−1, x21<x19<⋯<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≤−1, x is the largest. When −1<x<0, x21 is the largest. When 0≤x<1, x is the largest. When x≥1, x21 is the largest. Note that at x=0, x=0 and x21=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 x21), so it's not differentiable. At x=0, the left derivative is 0 (from x21) 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 x21), 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.