Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Verify that sec \(x\) has local minima at \(x=2 \pi k\) and local maxima at \(x=\pi+2 \pi k(k\) an integer) by identifying its critical points and using the second derivative test for maxima and minima.

Step-by-Step Solution

Answer

The function sec \(x\) has a local minimum at \(x=2 \pi k\) and a local maximum at \(x=\pi + 2 \pi k\) for every integer \(k\). This is confirmed by the second derivative test.

Step 1: Find the derivative of the function

The derivative of sec \(x\) is defined as sec \(x\) tan \(x\). Find the derivative to get the critical points.

Step 2: Find the critical points

The critical points occur when the derivative is undefined or when the derivative is equal to zero. For sec \(x\) tan \(x\), the critical points are when sec \(x\) = 0 and tan \(x\) = 0 which occur at \(x= 2 \pi k + \pi/2\) and \(x= \pi k\), respectively.

Step 3: Apply the second derivative test

The second derivative test states that if the second derivative at a point is positive, then the function has a local minimum at that point, and if the second derivative at a point is negative, then the function has a local maximum at that point. Find the second derivative of sec \(x\), which is sec \(x\) (sec \(x\) tan \(x\) + 2 tan^2\(x\)).

Step 4: Evaluate the second derivative at the critical points

Evaluate the second derivative at the critical points to determine whether these points are minima, maxima, or neither. For \(x=2 \pi k\), the second derivative is positive, indicating a local minimum. For \(x=\pi + 2 \pi k\), the second derivative is negative, indicating a local maximum.

Key Concepts

Critical Points

Critical points of a function are where its derivative equals zero or is undefined. These points are essential in calculus because they can signal where a function might change behavior, such as transitioning from increasing to decreasing. You start by finding the derivative of the function.
For the function secant, denoted as sec \(x\), the derivative is sec \(x\) tan \(x\).

• Set this derivative to zero or find where it is undefined to identify critical points.
• In our example, the critical points occur when either sec \(x = 0\) or tan \(x = 0\).
• However, sec \(x\) is never zero, so we focus on tan \(x = 0\).

This happens at multiples of \(\pi\), giving us critical points at \(x = \pi k\), where \(k\) is an integer.

Second Derivative Test

The second derivative test is a handy tool to determine the nature of critical points detected by the first derivative.
This test involves taking the second derivative of a function and evaluating it at the critical points discovered earlier.

• If the second derivative is positive at a given point, the function has a local minimum at that point.
• If it's negative, there's a local maximum.
• If the second derivative is zero, the test is inconclusive.

For sec \(x\), the second derivative is a bit more complex: \[ d^2/dx^2(secx) = sec \(x\) (sec \(x\) tan \(x\) + 2\tan^2(x) )\].
You'll evaluate this at the critical points to determine their nature.

Trigonometric Functions

Trigonometric functions like secant (sec \(x\)) and tangent (tan \(x\)) are periodic and have particular behaviors in calculus problems. These functions are based on the ratios of sides in right triangles but have extensive applications in various fields.

• The secant function is the reciprocal of the cosine function, defined as \(1/\cos(x)\).
• It is undefined wherever cosine is zero, at \(x = \pi/2 + n\pi\).
• Tangent, defined as \(\sin(x)/\cos(x)\), is zero where \(\sin(x)\) is zero, which occurs at integer multiples of \(\pi\).

Understanding these functions' behavior is pivotal for analyzing problems involving cycles or waves.

Local Maxima and Minima

Local maxima and minima refer to the highest or lowest points in a small neighborhood of a function. To find these, you look at the critical points and apply tests like the second derivative test.

• Local maxima are points where the function changes from increasing to decreasing.
• Local minima are where it changes from decreasing to increasing.
• The critical points offer potential locations for these extrema.

For sec \(x\), critical points at \(x = 2 \pi k\) are local minima since the second derivative is positive.
At \(x = \pi + 2 \pi k\), you find local maxima because the second derivative is negative at these points.