SOLUTION: Let f be a function such that f(0) = 2 and f'(x)<=6 for -10<=x<=10. A) What is the maximum possible value of f(4)? B) What is the maximum possible value of f(2)? C) Can f(5) b

Question 1027573: Let f be a function such that f(0) = 2 and f'(x)<=6 for -10<=x<=10.
A) What is the maximum possible value of f(4)?
B) What is the maximum possible value of f(2)?
C) Can f(5) be negative? Can f(5) = 0? Why or Why not?
D) Can f(5) = 31? Can f(5) = 35? Why or why not?
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
By the mean value theorem,
there is a such that
==> because for all x in [-10,10].
A) ==> ==> maximum possible value of f(4) is 26.
B) ==> ==> maximum possible value of f(2) is 14.
C) Can f(5) be negative? YES. The derivative in the neighborhood around x = 5 can be negative enough (and hence the graph fall drastically enough) so as to warrant the graph of of f(x) to cross the x-axis before x = 5. (Remember, f'(x) is less than or equal to 6 in [-10,10].)
Can f(5) = 0? YES, by a reasoning very similar to the preceding paragraph.
D) Can f(5) = 31? YES.
==> maximum possible value of f(5) is 32.
Can f(5) = 35? NO. The highest f(5) can get is 32
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