Question

Let
f(x) = x^2 if x ≤ 1
ax + b if x > 1.
Find the values of a and b so that f is continuous and has a derivative at x = 1.

Answer 1

Given the function
f(x)=x² x≤1
=ax+b x>2

For f(x) to be continuous at x=1, we need the following conditions:
1. f(1) exists and defined. Satisfied, f(1)=1.
2. Lim f(x) x->1- is defined and equal to f(1). Satisfied.
3. Lim f(x) x->1+ exists and equal to f(1). Depends on choice of a and b.

For f(x) to have a derivative at x=1, again we need the following conditions:
1. f'(1) exists and defined.
Satisfied, since f'(x)=2x, so f'(1)=2.
2. Lim f'(1) x->1- exists and equals f'(1). Satisfied.
3. Lim f'(1) x->1+ exists and equals f'(1) - to be satisfied by the proper choice of a and b.

Thus our choice of a and b are such that
slope=a=f'(1)=2
y=ax+b passes through (1,1).
so
(y-1)=2(x-1)
y=2x-1
which means a=2, b=-1.

See resulting function plotted graphically, curve applies to the left of x=1, and straight line applies to the right.

http://img163.imageshack.us/img163/3669/1296841563.png