Answer
The two solutions in the given interval are:
$x = 0$
$x = 0.3760$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/d82d95df-e334-4116-a209-c921ee7aa08f/result_image/1554702197.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240617%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240617T165900Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=5097e0371780a214ffd5fd76ec939f9c9f2b0c4df85011f73610113a2309c69f)
Work Step by Step
$x^3-cos^2~x = \frac{x}{2}-1$
$cos^2~x = x^3-\frac{x}{2}+1$
We can graph the two equations $~~y = cos^2~x~~$ and $~~y = x^3-\frac{x}{2}+1~~$ to find the points of intersection.
The blue graph is $~~y = x^3-\frac{x}{2}+1$
The red graph is $~~y = cos^2~x$
We can see that the points of intersection occur when $~~x = 0~~$ and $~~x = 0.3760$
The two solutions in the given interval are:
$x = 0$
$x = 0.3760$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/d82d95df-e334-4116-a209-c921ee7aa08f/steps_image/small_1554702197.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240617%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240617T165900Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=f70ec71b21a0caec4289fb52ba39a4d2e134bf91fe244511100704b57942b8a6)