Determining the Minimum Value of 100 sec(x) cos(x)

In this article, we will explore the process of determining the minimum value of the trigonometric function y 100 sec(x) cos(x). The function involves understanding the behavior of secant and cosine functions, their derivatives, and applying the second derivative test to find critical points and determine the nature of these points.

Introduction

The function y 100 sec(x) cos(x) is a complex one, involving secant (the reciprocal of cosine) and cosine functions. To find the minimum value of this function, we need to first determine its critical points and then use the second derivative test to classify these points.

Deriving the First Derivative

Let's start by finding the first derivative of the function y 100 sec(x) cos(x).

y 100 sec(x) cos(x)

Using the product rule of differentiation:

dy/dx d(100 sec(x)) / dx * (cos(x)) (sec(x)) * (100 d(cos(x)) / dx)

dy/dx 100 sec(x) tan(x) * cos(x) - 100 sec(x) sin(x)

Simplifying further:

dy/dx 100 sec(x) (tan(x) cos(x) - sin(x))

100 sec(x) (sin(x)/cos^2(x) - sin(x))

100 sec(x) (sin(x) - sin(x) cos^2(x) / cos^2(x))

100 sec(x) (sin(x) - sin(x) / cos^2(x))

100 sec(x) (x - sin^2(x))

Setting dy/dx 0 to find critical points:

100 sec(x) (sin(x) - sin^2(x)) 0

Two possibilities:

1. sec(x) 0

This is not possible as sec(x) 1/cos(x), and cos(x) ≠ 0 for real x.

2. sin(x) - sin^2(x) 0

sin(x)(1 - sin(x)) 0

sin(x) 0 or sin(x) 1

This gives us x 0°, 180°, and 90° (nπ for integers n).

Second Derivative Test

To determine whether these points are minima or maxima, we use the second derivative test. We start by finding the second derivative of the function:

d^2y/dx^2 d(dy/dx) / dx

Substituting the expression for dy/dx:

d^2y/dx^2 100 sec(x) d(d(sin(x) - sin^2(x))/dx) / dx

Using the chain rule and product rule:

d^2y/dx^2 100 sec(x) (sec(x) tan^2(x) - 2 sin(x) cos(x)

Substituting x 0°:

d^2y/dx^2 100 sec(0°) (sec^3(0°) - 0)

Since sec(0°) 1:

d^2y/dx^2 100 (1 - 0) 100 positive

This indicates that x 0° is a local minimum.

Conclusion

The minimum value of the function y 100 sec(x) cos(x) occurs at x 0°. Substituting x 0° into the original function:

y 100 sec(0°) cos(0°) 100(1)(1) 100

Thus, the least value of the function is 100.

Keywords

Minimum value Trigonometric functions Second derivative test