Find the x-value where the maximum value of y = sinx−2x^2 occurs.

Question

When you need to solve for an expression equal to zero, use Newton's method with x=0 as your first guess and enter your second guess as the answer. Enter your answer as a fraction.

Answer 1

This problem requires you to apply the 1st derivative test to the function y = sin(x) - 2x².

y ' = cos(x) - 4x

When does that equal zero?:

cos(x) - 4x = 0

This cannot be solved algebraically, so use Newton's Method to approximate the root(s) of:

g(x) = cos(x) - 4x

N(x) = x - [g(x)/g'(x)]

g'(x) = -sin(x) - 4 (note that g' is always negative, so g is always decreasing so there is only one root)

N(0) = 0 - [g(0)/g'(0)] = 0 - [1/-4] = 1/4 (this seems to be the value requested in the post).

You could get a better approximation:

N(1/4) = 1/4 - [g(1/4)/g'(1/4)] ≈ 0.24268...

Note that y'(0) is positive so, y is increasing to the left of 1/4.

y'(1) is negative, so y is decreasing the right of 1/4.

That means the 0.24268... x-value generates a relative maximum for the original function.

Here is a Desmos graph using Newton's Method to get a better approximation for the target x-value:

desmos.com/calculator/c47yasvy2o

x ≈ 0.242674680641