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

True, see proof below.

Step-by-step explanation:

Remember two theorems about continuity:

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.

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