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

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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.

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