Newtons Method help

Question

Find the x-value where the maximum value of  occurs. When you need to solve for an expression equal to zero, use Newton's method with  as your first guess and enter your second guess as the answer. Enter your answer as a fraction.

Answer 1

In order to find a relative maximum value of a function you will use the 1st derivative test, setting the first derivative equal to zero. In this case, let g(x) = sinx - 7x². Then g'(x) = cosx - 14x. To find the critical numbers set equal to zero and solve.

cos x - 14 x = 0 is not solvable algebraically, that is why Newton's Method is being suggested.

N(x) = x - f(x)/f'(x) where f(x) = cos x -14x. Let x₀=1 and find N(1) and call that x_1.

Continue in that way finding x₂ = N(x₁), x₃=N(x₂), .... until decimal digits start repeating. x₀ will change for different problems depending on the likely location of a root.

After you have tried the above, verify your results here:

Here is a graph depicting the graph of the original function and its 1st derivative:

desmos.com/calculator/nnkifl6tyj

And here is a graph that calculates a root using Newton's Method:

desmos.com/calculator/24dl4p7nnq

Scroll to the bottom of that 2nd graph to see the maximum value of the original function.