# Using Maple to numerically find the unique extremum (“turning point”) of a given function

Given $$f(x) = 5\sin\left(\frac14 x^4\right) -\sin\left(\frac12 x\right)^4$$ 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.

original image here

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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

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);

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

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