Question

if f is a differentiable function and f(0)=-1 and f(4)-3 then which of the following must be true there exists a c in [0,4] where f(c)=0

Answer 1

Answer:

True, see proof below.

Step-by-step explanation:

Remember two theorems about continuity:

1. If f is differentiable at the point p, then f is continuous at p. This also applies to intervals instead of points.
2. (Bolzano) If f is continuous in an interval [a,b] and there exists x,y∈[a,b] such that f(x)<0<f(y), then there exists some c∈[a,b] such that f(c)=0.

If f is differentiable in [0,4], then f is continuous in [0,4] (by 1). Now, f(0)=-1<0 and f(4)=3>0. Thus, we have the inequality f(0)<0<f(4). By Bolzano's theorem, there exists some c∈[0,4] such that f(c)=0.