We are asked to find the x-coordinates of the intersection(s) of the curves `y=1/4 x^2` and `y=4sin(x)-1`, or alternatively, the zeros of `f(x)=1/4 x^2-4sinx+1` .

There isn't a straightforward algebraic way to get the intersections. We can graph the two functions to estimate their intersections. (Alternatively graph f(x) and estimate the zeros.) We could then use guess and revise to narrow the answer to the required precision. (See attachment: `y=1/4x^2` is the green parabola, `y=4sin(x)+1` is the blue sinusoid, and f(x) is in red.)

We can use a graphing utility or program to approximate the zeros of f(x) or the intersections of the curves. My graphing calculator gives .25694421 and 2.4616646.

With some calculus, you can use Newton's method for approximating roots. (It works in this case. There are some curves for which the method fails.)

Choose a guess for the first intercept; say x=0. Then, with `x_0=0`, we calculate `x_1=x_0-(f(x_0))/(f'(x_0))` where `f(x)=1/4x^2-4sinx+1` and `f'(x)=1/2x-4cosx` .

So, `x_0=0,x_1=-1/4,x_2~~.2512287988,x_3~~.2569376661` which is accurate to 4 decimal places.

Choosing 2.5 for the other intercept yields the sequence 2.5,2.462148702,2.461665119.

We can be sure that there are only two intersections. `y=4sinx -1` is bounded; `-5<=y<=3` . But, `y=1/4x^2>3` whenever `|x|>2sqrt(3)` . So, we need only check the interval `[-2sqrt(3),2sqrt(3)]` and `y=4sinx-1<0` for x<0 in this interval.

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