Problem 12 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the maximum and minimum values of \(\cos 2 x+9 \sin x\).

Step-by-Step Solution

Answer

The exact minimum and maximum values will depend on numerical solutions for the critical points obtained from the equation \(-2\sin 2x + 9\cos x = 0\). However, the minimum and maximum will always lie at these critical points manifesting themselves when the function changes direction (from increasing to decreasing for maximum points, and from decreasing to increasing for minimum points).

Step 1: Differentiate the function

Calculate the first derivative of the function \(f(x) = \cos 2x + 9 \sin x\). Using the chain rule, we'll have \(f'(x) = -2\sin 2x + 9\cos x\). This gives the slope of the function at any point x.

Step 2: Find the critical points

Set the first derivative equal to zero and solve for x: \(-2\sin 2x + 9\cos x = 0\). This transcendental equation doesn't have an expression for its roots, so it requires numerical methods or a graphing calculator to find the solutions. The solutions to this equation are the critical points.

Step 3: Analyze the interval between critical points

By plugging in values from each interval between the critical points into the derivative, we can determine whether the function is increasing or decreasing on that interval. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.

Step 4: Determine the Maximum and Minimum Values

Where the function changes from increasing to decreasing at a critical point, we have a local maximum. Where the function changes from decreasing to increasing at a critical point, we have a local minimum. Considering the periodic nature of \(f(x)\), we must also analyze the boundaries of the period to assure we have the absolute maximum/minimum points.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in calculus, especially when it comes to optimizing functions that involve periodic behavior. In the function given, \(\cos 2x + 9\sin x\), both \sin x\ and \cos 2x\ are trigonometric expressions. The function combines these to model a new periodic pattern.
These trigonometric components have specific properties:

• They are periodic, meaning they repeat their values over regular intervals.
• \sin(x)\ and \cos(x)\ both have a range between -1 and 1.

Thus, any function involving sine or cosine will oscillate between certain bounds based on these ranges. Understanding this helps with anticipating the behavior of the function, which is crucial when determining points of maximum or minimum value.

Critical Points

Critical points in calculus are where the derivative of the function is zero or undefined. They are essential for finding where a function identifies potential peaks and troughs. For our function, finding critical points involves taking the derivative, \( f'(x) = -2\sin 2x + 9\cos x \), and setting it to zero.

• This results in the equation \-2\sin 2x + 9\cos x = 0\.
• To solve for \x\, numerical methods or graphing calculators are commonly used because of the complexity of the trigonometric equation.
• These solutions are the critical points of the function, indicating where potential changes in the direction of the slope occur.

Identifying these points is an integral part of analyzing the behavior of the function in the domain, particularly for determining maxima and minima.

Maxima and Minima

Determining maxima and minima of a function involves analyzing where the function reaches its highest and lowest values, either locally or globally. After locating the critical points of a function, as discussed, it's time to identify if these points are maxima, minima, or neither.

• We determine this by observing the sign change of the derivative around these points.
• If the derivative changes from positive to negative at a critical point, that point is a local maximum.
• If the derivative changes from negative to positive, it's a local minimum.

The function \(\cos 2x + 9\sin x\) is periodic, so analyzing within its period is crucial. The same critical point analysis applies, yet checking the entire period ensures any absolute maximum or minimum is accurately identified.

Chain Rule

The Chain Rule is a fundamental tool in calculus for differentiating composite functions, such as functions composed of two or more simple functions. In the context of the function \( \cos 2x + 9\sin x \), the Chain Rule helps find the derivative accurately.

• For \(\cos 2x\), the outer function is cosine, and the inner function is \(2x\). The derivative of \cos(2x)\ applies the Chain Rule to yield \-2\sin(2x)\.
• Each part of the function is treated systematically to account for inner transformations within the trigonometric function.

By using the Chain Rule, we can build the correct derivative, enabling the next step in finding critical points. This derivative lays the groundwork for subsequent analysis of the function's behavior.