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

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
By the mean value theorem,
there is a 0%3C=+alpha+%3C=+x such that f%28x%29+=+f%280%29+%2B+%28df%28alpha%29%2Fdx%29x
==> f%28x%29+%3C=+f%280%29+%2B+6x+=+2+=+6x because df%28x%29%2Fdx+%3C=+6 for all x in [-10,10].
A) ==> f%284%29+%3C=+2+%2B+6%2A4+=+26 ==> maximum possible value of f(4) is 26.
B) ==> f%282%29+%3C=+2+%2B+6%2A2+=+14 ==> 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.
f%285%29+%3C=+2+%2B+6%2A5+=+32 ==> maximum possible value of f(5) is 32.
Can f(5) = 35? NO. The highest f(5) can get is 32