determine the differentiability of the funtion f(x)= {x+1, for x<0 and x^2+1, for x>=0}

Answer 1

To determine the differentiability of the function f(x), we need to check if the function is continuous and if its derivative exists at the point of interest.

First, let's check for continuity at x=0.

Since f(x) is piecewise, we need to check the limits from the left and right of x=0.

For x < 0:
lim (x->0-) f(x) = lim (x->0-) (x + 1) = 1

For x >= 0:
lim (x->0+) f(x) = lim (x->0+) (x^2 + 1) = 1

Since the limits from the left and right are equal at x=0, f(x) is continuous at x=0.

Next, let's check the derivative at x=0.

For x < 0:
f'(x) = d/dx (x + 1) = 1

For x >= 0:
f'(x) = d/dx (x^2 + 1) = 2x

Thus, at x=0, f'(0) = 0.

Since the function is continuous and the derivative exists at x=0, f(x) is differentiable at x=0.