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https://mapleta.telt.unsw.edu.au:8443/mapleta/tmp/ih/dk/mh/galofgccadnjemkkocleeenjbh.gif

Find, to 10 significant figures, the unique turning point of x[0] in the interval [1,2]. Also, i've got to get the second derivative in 10 significant figures.

The plot doesn't exactly make sense in maple, don't think i understand how to go about solving it mathematically.

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Also, you will need the definition of "turning point" and the connection between $x$ and $\mathtt{x[0]}$. –  GEdgar Sep 26 '12 at 17:30
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2 Answers 2

$\tt fsolve(diff(5*sin(x^4/4)-sin(x/2)^4, x\$2), x, 1 .. 2)$ gives 1.321411467.

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If by "turning point" you mean relative extrema, then the only relative max on your interval is located at $x \approx 1.576726466$.

[> f := x -> 5*sin(x^4/4)-sin(x/2)^4;
[> plot(f(x),x=1..2);

plot of f(x)

Setting $f'(x)=0$ and solving we get...

[> x[0] := fsolve(diff(f(x),x)=0,x=1.6);

                      1.576726466

Plugging this into the second derivative we get...

[> evalf(subs(x=x[0],diff(f(x),x,x)));

                      -76.34072337
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