Finding the Minimum and Maximum Values of a Function Involving Inverse Trigonometric Functions

Understanding how to find the minimum and maximum values of a function, especially one involving inverse trigonometric functions, can be essential for various applications in mathematics and science. One such example is the function f(x) arcsin(x) * arccos(x3). This article will guide you through the process, providing a detailed step-by-step explanation to help you grasp the concepts fully.

Defining the Domain and Rewriting the Function

First, it's important to establish the domain of the function f(x) arcsin(x) * arccos(x3). Both arcsin(x) and arccos(x) are defined for x ∈ [-1, 1]. This means that the domain of f(x) is also limited to x ∈ [-1, 1].

We can simplify the function using the identity arcsin(x) * arccos(x) π/2. Applying this identity, we rewrite the function as:

f(x) arcsin(x) * (π/2 - arcsin(x)3)

Expressing in Terms of a New Variable

Let's introduce y arcsin(x). This implies that x sin(y) and the range of y is [-π/2, π/2]. Rewriting the function in terms of y, we have:

f(y) y * (π/2 - y)3

Differentiating to Find Critical Points

To find the minimum and maximum values, we need to differentiate f(y) with respect to y and set the derivative equal to zero:

f'(y) 1 - 3(π/2 - y)2

Setting f'(y) 0, we solve for y:

1 - 3(π/2 - y)2 0

Simplifying, we get:

(π/2 - y)2 1/3

Solving for y, we obtain two solutions:

y π/2 ± 1/√3

Evaluating at the Critical Points and Endpoints

We need to evaluate f(y) at the critical points and the endpoints of the interval [-π/2, π/2].

When y -π/2:

f(-π/2) -π/2 * (π/2 π/2)3 -π/2 * π3

When y π/2:

f(π/2) π/2 * (π/2 - π/2)3 π/2 * 0 π/2

Since f(y) is strictly increasing on the interval [-π/2, π/2], the minimum value occurs at y -π/2, and the maximum value occurs at y π/2.

Conclusion

Thus, the minimum and maximum values of the function f(x) arcsin(x) * arccos(x3) are:

This step-by-step approach helps to clearly identify and compute the critical points and endpoints, providing a solid understanding of how to determine the minimum and maximum values of such functions involving inverse trigonometric operations.