Continuity and Differentiability · Question 16

Let m and n be the number of points at which the function f(x) = \x, x3, x5, , x21 \, x R, is not differentiable and not continuous, respectively. Then m + n is equal to .

Answer

3

Solution

All the functions x, x3, , x21 are odd powers, continuous everywhere.

x > 1: higher power is larger f(x) = x21

0 x 1: lower power is larger f(x) = x

-1 x 0: all values are 0 and x21 is nearest to 0 f(x) = x21

x < -1: x is the least negative f(x) = x

f is a maximum of continuous functions, so f is continuous everywhere n = 0

At x = -1: slopes 1 (left) and 21 (right); at x = 0: slopes 0 and 1; at x = 1: slopes 1 and 21.

f is not differentiable at x = -1, 0, 1 m = 3

m + n = 3

Answer: 3