Question

Golden ratio method is recommended by the solution and the numerical solution is 4.493335656115932 (or anything accurate up to 2 dp)

(2) Maximum of a function We consider the function, f(x) = ( Sinar Provided that there is a local maximum in the interval r 4,5. Please find this local maximum with accuracy up to two decimal places. You need to prove that the desired accuracy is reached so that you can stop. You can use any methods you know.

Answer 1

$\small f(x) = \left(\frac{sinx}{x}\right)^{2}$

$\small f'(x) = x\cos x - \sin x$

For maxima f'(x) must be zero.

$\small x\cos x - \sin x = 0\; \; \; \; \; \; ;\; \; \; \; \; x\epsilon [4,5]$

Let us use the bisection method, (I'm using f(x) instead of f '(x) from here on)

4+5 First guess value,11 =-= 4.5 f(x) 4.5 * cos 4.5-sin 4.5 0.02894902022 > 0 Second guess value,x2= f(x2) 4.25 * cos 4.25-sin 4.25 =-1 .00088247391 < 0 I hird guess ualue,T3 f(r3)-4.375 cos 4.375-sin 4.3750.50460958943 < 0 4.5+4-4.2 4.5 + 4-25-4375

4.375+ 4.5 4.4375 Fourth guess value,14 f(4.4375* cos4.4375 -sin4.4375-0.24206004825 0 Fifth guess value,15 = ()0.0217040656 < 4.4375+4.4.46875 4.4875+4.54.484375 f(r6)0.03954592614

4.484375 +4.5_44021875 17 f(17) 4.4921875+4.5 = 4.49609375 f(28) 78 -一0.0053581 7723 < 0 0.01178060816>0 4.49609375 4.4921875 = 4.494140625 0.00320749737 > 0 f(m)

Proceeding further will guide you to the exact value of the answer. You can also use other numerical techniques.

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By plotting the function f '(x) also one can find the solution

10 10 5 5 0 0 5 10

above is the graphical representation of f '(x)