Determine whether the statement is true or false:a)If f(1)>0 and f(3)<0, then there exists a number c between 1 and 3 such that f(c)=0 b)If f is continuous at a, then f is differentiable at a.

Answer 1

The easiest way to disprove is with counterexamples.

a) False; let f(x) = 1 x ≠3 and f(3) = -1.

This function has f(1) > 0, f(3) < 0, and no c for which f(c) = 0. The requirement missing for f(c) = 0 for some x between 1 and 3 is that f(x) is not continuous.

b)False; let f(x) = |x|

This function is continuous everywhere, but it is not differentiable at x = 0