Conclusions from derivative

Asked Feb 25 '14 at 04:33

Active Feb 25 '14 at 04:43

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Suppose f is continuously differentiable on [a,b] and f(a)=2 and $|f'(x) \leq 0.3|$ for all x. What can you say about f(b)?

The only result I could get was $\frac{f(b)-2}{b-a} \leq 0.3$ by the mean value theorem, which is not insighful. What else can be done here?

asked Feb 25 '14 at 04:33

clickclock

hint: try rearranging and finding out what $f(b)$ could be. – Newb Feb 25 '14 at 04:37
Hmm, what I have here is 3 unknowns, $f(b) \leq 0.3(b-a)+2$ – clickclock Feb 25 '14 at 04:39
You can find an interval where $f(b)$ lies in it. – Mhenni Benghorbal Feb 25 '14 at 04:40
Is the expression I have above the right bound? How would I go from this to an interval? – clickclock Feb 25 '14 at 04:42
It is not the right bound. We need absolute value signs. – André Nicolas Feb 25 '14 at 04:54
2

From the MVT we have $\frac{|f(b)-2|}{b-a}\le 0.3$. This can be rewritten as $2-(b-a)(0.3)\le f(b)\le 2+(b-a)(0.3)$. – André Nicolas Feb 25 '14 at 05:22

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