If f(4) = 4 and f '(x) geq 2 for 4 leq x leq 7, how small can f(7) possibly be?

Question
Functions
asked 2021-02-23
If \(f(4) = 4\) and \(f '(x) \geq 2\) for \(4 \leq x \leq 7\), how small can f(7) possibly be?

Answers (1)

2021-02-24
Assuming that the function is continuous on the interval [4,7][4,7], you could use the mean value theorem.
For \(x∈(4,7)x∈(4,7)\), one has by the mean value theorem that :
\(f'(x)=\frac{(7)-4}{3}\geq2\)
So that
\(f(7)\geq10\)
ie the smallest value that f(7) can obtain is 10.
0

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