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I have a function

[tex] f(x) = \frac{\cos(x^2)+x}{x^2+2},\quad x\in[0,3] [/tex]

I have to find the extreme values of the function in in the range [0,3], with Maple, by solving f'(x) = 0. Maple will solve these numerical, and I get 3 values.

[itex]c_1 = 0.5345058769[/itex], [itex]c_2=1.732313261[/itex] and [itex]c_3=2.461303848 [/itex].

Now there is an uncertantity in this, which can be seen, by calculating f'(c), for c1 (which should be a maxima) it is f'(c1) = -2*10^(-10). Surly this value x = c1 most be a little to the right of the true value of the maxima. Now how can I confirm that there isn't anyvalues in a small range around c1, so that [itex] f(c_1-\delta) \gg f(c_1) [/itex] for a very small value of [itex]\delta>0[/itex]?

How can I use elementary Calculus rules/theorems to argument about this?

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# Function extrema and Maple accuracy

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