Suppose #f# is a differentiable function such that #f(2)=-4# and #f'(x)<=3# for all #x#. What is the maximum value that #f(5)# could be?

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Answer 1 · 2017-04-22T01:05:28

The maximum it could be is 5.

The inequality, #f'(x)<=3#, tells us that 3 is the maximum slope that the line connecting f(2) with f(5) could have:

#3 = (f(5) - f(2))/(5-2)#

Substitute -4 for #f(2)# and perform the subtraction in the denominator:

#3 = (f(5) - -4)/(3)#

Solve for maximum of #f(5)#:

#9 = f(5) - -4#

#f(5) =5#