How to Find the Extreme Points of y [Cos xa Sin x] / [Cos^2 a]

Understanding the behavior of trigonometric functions with parameters is crucial in various mathematical and engineering applications. This guide will demonstrate how to find the extreme points of the function y (Cos xa Sin x) / (Cos^2 a), where x is a variable and a is a parameter. We will also discuss how to enter and solve the function using Mathematica.

Step-by-Step Analysis

Deriving the Critical Points

To find the extreme points of the function, we first need to find its derivative with respect to x:

y' (1 / Cos a2) - (Sin xa Sin x Cos xa Cos x) 0

Using trigonometric identities, this can be simplified to:

Sin x Sin xa Cos x Cos xa

Further simplification gives:

Tan x Cot xa

Thus, x 45° 180k - a/2, where k is an integer. We can further refine this to the general form:

x 45° 180k - a/2

By analyzing the sign of the second derivative, we can determine whether these points are maxima or minima. Generally, even values of k correspond to maxima, and odd values to minima.

Using

Mathematica Code Implementation

Mathematica is a powerful tool for symbolic and numerical computations. Here is how you can enter and differentiate the given function to find its critical points:

Step 1: Define the function in Mathematica

y[x_, a_] : (Cos[x*a]*Sin[x])/Cos[a]^2

Step 2: Find the derivative of the function

yPrime[x_, a_]  D[y[x, a], x]

Step 3: Solve for critical points

criticalPoints  Solve[yPrime[x, a]  0, x]

The Solve function will return a list of solutions, including both maxima and minima. To distinguish between maxima and minima, you can analyze the second derivative at these points.

Step 4: Determine the nature of the critical points

To determine whether the points are maxima or minima, we can compute the second derivative:

ySecondDerivative[x_, a_]  D[yPrime[x, a], x]

Then we can use the second derivative test:

maxima  Select[criticalPoints, ySecondDerivative[#[[1]], a]  0 ]minima  Select[criticalPoints, ySecondDerivative[#[[1]], a]  0 ]

Step 5: Calculate the specific values of the function at maxima and minima

Finally, to find the specific values of the function at these points, you can substitute the critical points back into the original function:

maximaValues  y[#[[1]], a]  /@ maximaminimaValues  y[#[[1]], a]  /@ minima

Conclusion

By following these steps, you can find the extreme points of the function y (Cos xa Sin x) / (Cos^2 a) and determine whether they are maxima or minima. Mathematica provides a robust and convenient platform for symbolic and numerical computations, making it an invaluable tool for such tasks.