continuous and differentiable function

Question

Let f be a continous function on interval [-1, 9] and differentiable on (-1, 9).  If f(-1) = 2 and f(9) = 7, then which of the following are necessarily true?

 

i.   f'(c)= (1/2) for some c between -1 and 9

 

ii.  f(c)>0 for all c between -1 and 9

 

iii.  f(c)=5 for some c between -1 and 9

 

 

A.) I. only             
B.) II. only                        
C.) I. and II. only
D.) I. and III. only
E.) I., II., and III.

 

Answer 1

Since f is continuous on [-1, 9] and is differentiable on (-1, 9), the Mean Value Theorem applies.  By the Mean Value Theorem, there is at least one number, c, between -1 and 9 such that

 

f'(c) = [f(9)-f(-1)]/[9-(-1)] = 5/10 = 1/2

 

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Since f(x) is continuous on the interval [-1, 9] and f(-1) = 2 < 5 and f(9) = 7 > 5, then by the Intermediate Value Theorem, f(c) = 5 for at least one number, c, between -1 and 9.

 

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The graph of f could dip below the x-axis somewhere between

x = -1 and x = 9. So we can not conclude that f(c) > 0 for all c between -1 and 9.  

 

ANSWER:  (D). only I and III are necessarily true