Question

Use Newton's method to find the absolute maximum value of the function f(x) = 8x sin(x), 0 ≤ x ≤ π correct to SIX decimal places.

Answer 1

We know that at The maximum point f '(x)=0
So using Newton's method, the derivative becomes the function, and the derivative is f ''(x)

On differentiating,we get

g(x) = f '(x)= (8x)cosx + 8sinx= 0
g '(x)= (8x)(-sinx)+ 8cosx + 8cosx = -8x sinx +16cosx

Graph of f(x) =8x sinx

Graph of g(x)=8sinx+8x cosx

Looking at a graph, the first guess should be around x=2

X1= 2

X2= 2-g(2)/g''(2)= 2- (.61603)/(-21.21)= 2.0290483

X3= 2.0290483- g(2.0290483)/g'(2.0290483)= 2.0287579

X4= 2.0287578

This is accurate to six places, since x3= x4= 2.028758

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