In each case justify your answer by stating a theorem showing the statement is t
ID: 2841918 • Letter: I
Question
In each case justify your answer by stating a theorem showing the statement is true, or drawing a garph of a function for which the statement is false.
a. If f(x) is continuous on the interval [-1,1] and if f(-1)=f(1)=1, then there must be a number c in [-1,1] where f'(c)=0.
b. If f(x) is a polynomial and if f(x)=2 and f(2)=0, then there is a number c in [0,2] where f'(c)=-1.
c. If f(x) is differentiable at x=3 and if f'(3)=0 then f has a local maximum or a local minimum at x=3.
d. If f'(x) exists and f'(x)>0 at every x in interval (3,6), then f(x) is increasing in the interval (3,6).
e. If f'(3)=0 and if f''(3)=1, then f has a local maximum at x=3.
Explanation / Answer
a) True
b) True
a and b is based on LMVT If f(x) is continuous on the interval [-1,1] and if f(-1)=f(1)=1, then there must be a number c in [-1,1] where f'(c)=0.
c) False (could be a point of inflection)
d)true (it's characteristic of a increasing function)
e) false (it's minima not maxima)
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