Maximum value of endpoint

Calculus Level 2

Suppose a function f : [ 0 , 10 ] → R f:[0,10] \rightarrow \mathbb{R} is continuous and differentiable everywhere in its domain. If f ( 10 ) = 19 f(10)=19 , and ∣ f ′ ( x ) − 5 ∣ ≤ 4 \mid f'(x)-5 \mid \leq 4 for all x x in the domain, what is the maximum value of f ( 0 ) f(0) ?

-71 19 1 9

Charley Shi by Charley Shi , 19, New Zealand

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2 solutions

2 Helpful 0 Interesting 2 Brilliant 0 Confused
Chris Lewis
Apr 16, 2019

In order to maximise f ( 0 ) f(0) , we want f ′ ( x ) f'(x) to be as small as possible across the whole domain. So we simply set f ′ ( x ) = 1 f'(x)=1 (which is the smallest value allowed). Using the fact that f ( 10 ) = 19 f(10)=19 , we get f ( x ) = x + 9 f(x)=x+9 , and so f ( 0 ) = 9 f(0)=\boxed9 .

3 Helpful 1 Interesting 1 Brilliant 0 Confused

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