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The First Derivative Test is a handy tool to determine whether a critical point of a function is a local maximum or minimum. Critical points occur where the derivative of the function either equals zero or is undefined. To apply the test effectively, you need to:

• Identify critical points by setting the first derivative equal to zero.
• Check the sign of the derivative before and after each critical point.

For the function \( f(\theta) = \sin 2\theta \), the first derivative is \( f'(\theta) = 2\cos 2\theta \). By solving \( 2\cos 2\theta = 0 \), we look for values where the derivative changes sign.
In our specific case, there are no true critical points within the interval \( 0 < \theta < \frac{\pi}{4} \). To gain insight, it's helpful to evaluate the derivative's sign as \( \theta \) approaches the endpoints of the interval. At \( \theta = 0^+ \), the derivative is positive, indicating an increasing function. Near \( \theta = \frac{\pi}{4}^- \), it is still positive, suggesting a local maximum as \( \theta \) approaches \( \frac{\pi}{4} \).

The Second Derivative Test provides further confirmation of the nature of critical points by examining concavity. Once you have a potential critical point, you can:

• Find the second derivative of the function.
• Evaluate this second derivative at the critical point.
• Determine the concavity to see if the point is a maximum or minimum.

For \( f(\theta) = \sin 2\theta \), the second derivative is \( f''(\theta) = -4\sin 2\theta \). This test is useful especially at the boundary points of intervals, where the First Derivative Test might be inconclusive.
Evaluating the second derivative, we find that at \( \theta = \frac{\pi}{4} \), \( f''(\theta) = -4 \). This negative value indicates the function is concave down, confirming that \( \theta \to \left(\frac{\pi}{4}\right)^- \) is indeed a local maximum, as a negative second derivative suggests a peak or a maximum point.

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When you have a function nested within another function, the chain rule allows you to find the derivative systematically.

• Identify the outer function and the inner function.
• Differentiate the outer function while keeping the inner function unchanged.
• Multiply the result by the derivative of the inner function.

In our problem, the function \( f(\theta) = \sin 2\theta \) is a perfect candidate for the chain rule because it involves the composite function \( \sin(x) \) and \( 2\theta \). First, \( \sin x \) is the outer function, and \( 2\theta \) is the inner function. Differentiating gives us:\[ f'(\theta) = 2 \cos 2\theta \], where the chain rule helps to multiply the derivative of the outer function \( \cos x \) by the derivative of the inner function \( 2 \). The chain rule is powerful for handling such nested compositions and is a cornerstone of finding derivatives elegantly.