Calculus 1 Help!!?

Question

I'm totally lost on these problems- please help if you can!!

1. Show that there is some number "c" with 0
2. For what real value(s) of b will the piecewise function g(x)= x, if x<0 and bx^2, if x is greater than or equal to 0
be continuous at every x in R?

Answer 1

#1: Consider that f(0) = 0-1 = -1 < 0, whereas f(π/2) = π/2 - 0 = π/2 > 0. Since f(0)<0, f(π/2)>0, and f is continuous, ∃c∈(0, π/2) s.t. f(c) = 0. Then c - cos c = 0, so c = cos c.

As a side note, you can find the value of c simply by turning on windows calculator, setting it to radians, and hitting the cosine button over and over until the display stops changing. You find that c ≈ 0.73908 51332 15160 64165 53120 87673 87

#2: Obviously the function is continuous at every point except 0, and for it to be continuous at 0, the left-hand limit must be equal to the right-hand limit, so in particular [x→0⁺]lim g(x) = [x→0⁺]lim bx² = b*0² = 0 must equal [x→0⁻]lim g(x) = [x→0⁻]lim x = 0. But 0 always equals 0, so this is continuous for every value of b.