Answer
On the graph, we can see that the maximum value of $\theta$ occurs when $x \approx 1.4$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/d1d9e5ce-45e1-46bd-bfdc-6bc1774adf22/result_image/1554643870.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T001159Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=fee7641e3ee46e8265ba3181180ddf3a6b8f388a74a76b813966d66bfe3c193f)
Work Step by Step
We can graph the following function:
$\theta = tan^{-1}(\frac{x}{x^2+2})$
On the graph, we can see that the maximum value of $\theta$ occurs when $x \approx 1.4$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/d1d9e5ce-45e1-46bd-bfdc-6bc1774adf22/steps_image/small_1554643870.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T001159Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=6c70fc016f1acfd013f18ff79e520c12f9c783c3e993085fff3545ced1dcaeb1)