We are tasked with finding the value of $x$ that maximizes the function:

$f(x) = 150 \cos(x) \left( 15 \sin(x) + \sqrt{224 \sin^2(x) + 20} \right)$

Step 1: Simplify the function

Let:

$A = 150 \cos(x), \quad B = 15 \sin(x) + \sqrt{224 \sin^2(x) + 20}$

So, the function becomes:

$f(x) = A \cdot B$

Where:

$A = 150 \cos(x), \quad B = 15 \sin(x) + \sqrt{224 \sin^2(x) + 20}$

Step 2: Take the derivative

We need to differentiate $f(x)$ with respect to $x$. Using the product rule:

$\frac{d}{dx} \left( A \cdot B \right) = A' \cdot B + A \cdot B'$

Now, we compute the derivatives of $A$ and $B$:

• The derivative of $A = 150 \cos(x)$ is:

$A' = -150 \sin(x)$

• The derivative of $B = 15 \sin(x) + \sqrt{224 \sin^2(x) + 20}$:

  We differentiate the two terms separately:

  • The derivative of $15 \sin(x)$ is $15 \cos(x)$.
  • For the second term $\sqrt{224 \sin^2(x) + 20}$, use the chain rule. Let:

  $g(x) = 224 \sin^2(x) + 20$

  Then:

  $\frac{d}{dx} \left( \sqrt{g(x)} \right) = \frac{1}{2 \sqrt{g(x)}} \cdot \frac{d}{dx} \left( g(x) \right)$

  Now, compute the derivative of $g(x)$:

  $\frac{d}{dx} \left( 224 \sin^2(x) \right) = 448 \sin(x) \cos(x)$

  Thus, the derivative of the second term is:

  $\frac{224 \sin(x) \cos(x)}{\sqrt{224 \sin^2(x) + 20}}$

So:

$B' = 15 \cos(x) + \frac{224 \sin(x) \cos(x)}{\sqrt{224 \sin^2(x) + 20}}$

Step 3: Solve for critical points

To find the critical points, set the derivative $f'(x)$ equal to zero:

$-150 \sin(x) \left( 15 \sin(x) + \sqrt{224 \sin^2(x) + 20} \right) + 150 \cos(x) \left( 15 \cos(x) + \frac{224 \sin(x) \cos(x)}{\sqrt{224 \sin^2(x) + 20}} \right) = 0$

This is a complex equation, so solving it exactly may require numerical methods. However, we can analyze the behavior of $f(x)$ by inspecting certain key points of $\sin(x)$ and $\cos(x)$, such as $x = 0$, $\frac{\pi}{2}$, and $\pi$, where sine and cosine take on their maximum and minimum values.

Step 4: Analyze key points

We can evaluate the function $f(x)$ at key points where trigonometric values are well known to narrow down where the maximum might occur.

Let me calculate the specific values for these key points to determine where the maximum occurs.Evaluating the function $f(x)$ at key points:

• $f(0) = 670.82$
• $f\left(\frac{\pi}{2}\right) \approx 0$
• $f(\pi) = -670.82$

The maximum value occurs at $x = 0$, where $f(x) = 670.82$.

Conclusion:

The function reaches its maximum value when $x = 0$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the result change if the coefficients in the function were different?
2. What are the critical points of this function between $0$ and $2\pi$?
3. How does the behavior of sine and cosine functions affect optimization problems like this?
4. Can the function $f(x)$ have more than one local maximum in a different interval?
5. How do we apply numerical methods to solve more complex derivatives?

Tip: When optimizing trigonometric functions, always consider key points like $0, \frac{\pi}{2}, \pi$, etc., where sine and cosine take extreme values.