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Chapter 3: Problem 26

Finding Points of Inflection In Exercises \(15-30\) , find the points of inflection and discuss the concavity of the graph of the function. $$ f(x)=2 \csc \frac{3 x}{2}, \quad(0,2 \pi) $$

Short Answer

Expert verified
The points of inflection of the function \( f(x) = 2 \csc(\frac{3x}{2}) \) are \( x = \frac{2\pi}{3}, \frac{4\pi}{3} \). The function is concave down on the intervals \( (0, 2\pi/3) \) and \( (4\pi/3, 2\pi) \), and concave up on the interval \( (2\pi/3, 4\pi/3) \).

Step by step solution

01

Compute the first derivative

The first derivative of the function can be found by applying the quotient rule of derivatives. We get: \( f'(x) = \frac{d}{dx} [2 \csc(\frac{3x}{2})] = -3 \cdot 2 \cdot \csc(\frac{3x}{2}) \cdot cot(\frac{3x}{2})=-6 \cdot \csc(\frac{3x}{2}) \cdot cot(\frac{3x}{2}) \)
02

Compute the second derivative

The second derivative of the function is the derivative of the first derivative, which requires product rule and chain rule. We finally get: \( f''(x) = 9 \cdot 2 \cdot cot^2(\frac{3x}{2}) \cdot \csc(\frac{3x}{2}) - 3 \cdot 2 \cdot \csc^2(\frac{3x}{2}) = 18 \cdot cot^2(\frac{3x}{2}) \cdot \csc(\frac{3x}{2}) - 6 \cdot \csc^2(\frac{3x}{2}) \)
03

Find critical points

The points of inflection occur where the second derivative is equal to zero or undefined. We set \( f''(x) = 0 \) i.e. \( 18 \cdot cot^2(\frac{3x}{2}) \cdot \csc(\frac{3x}{2}) - 6 \cdot \csc^2(\frac{3x}{2}) = 0 \) and solve for \( x \). Also, we determine where \( f''(x) \) is undefined, which for \( cot(x) \cdot \csc(x) \) are multiples of \( \pi \). Because of our domain, the potential points are \( x = \frac{2\pi}{3}, \frac{4\pi}{3} \).
04

Determine concavity

Determine the concavity of \( f(x) \) on the interval \( (0,2\pi) \) by choosing test points in each subinterval determined by the critical points. Let's take \( \pi/2, \pi \) and \( 3\pi/2 \) as test points. Calculating \( f''(x) \) at these points will provide the concavity. If \( f''(x) > 0 \), the function is concave up on that interval; if \( f''(x) < 0 \), the function is concave down. After computation, we get that the function is concave down on \( (0, 2\pi/3) \) and \( (4\pi/3, 2\pi) \), and concave up on \( (2\pi/3, 4\pi/3) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Concavity
Concavity is a property of a graph that describes how the curve bends. When analyzing concavity, we focus on whether the function's graph is bending upwards or downwards between points. To determine this, we use the second derivative of the function. Here's how it works:

  • A function is concave up on an interval if its graph looks like a cup facing upwards. In simpler terms, if you find that the second derivative, denoted as \( f''(x) \), is greater than zero over this interval, then the function is concave up.
  • Conversely, a function is concave down if its graph forms an upside-down cup shape. This occurs when the second derivative is less than zero.

By identifying intervals of concavity, we can better understand the overall shape of the graph and identify points of inflection, where the concavity changes direction.
First Derivative
The first derivative of a function provides insight into the rate of change of the function itself. This derivative, often denoted as \( f'(x) \), tells us how the function's output changes as its input changes.

  • If \( f'(x) > 0 \), the function is increasing, meaning its graph moves upwards as you move from left to right.
  • If \( f'(x) < 0 \), the function is decreasing, which means the graph is moving downwards.
  • If \( f'(x) = 0 \), this typically indicates a stationary point, which could be a minimum, maximum, or a point of inflection.

In the given exercise, finding the first derivative involved applying the chain rule and recognizing terms like \( \csc(x) \) and \( \cot(x) \) that are interrelated.
Second Derivative
The second derivative, \( f''(x) \), is crucial in determining concavity and identifying points of inflection, where the graph changes from concave up to concave down or vice versa.

  • To find the second derivative, we differentiate the first derivative using appropriate rules, such as the chain rule and product rule, especially when functions involve trigonometric components like \( \csc(x) \) and \( \cot(x) \).
  • Critical points arise when the second derivative equals zero or is undefined. This helps in pinpointing potential points of inflection and understanding overall graph behavior.

In the exercise, calculating \( f''(x) \) was paramount to understanding where and how the graph's concavity shifts, helping determine these critical intervals.
Cotangent
The cotangent function, \( \cot(x) \), is the reciprocal of the tangent function and helps to create relationships between the sides of a right triangle. It is given by the formula:

\[ \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \]

Functions like \( \cot(x) \) often arise in calculus problems involving periodic behavior, and they have vertical asymptotes when the \( \sin(x) \) is zero. In derivatives and graph behavior analysis:

  • Cotangent can create vertical asymptotes, generating places where the function is undefined, which has implications for finding points of inflection.
  • In conjunction with cosecant, cotangent helps determine undefined regions for functions, an important step in calculating derivatives.
Cosecant
The cosecant function, denoted by \( \csc(x) \), is the reciprocal of the sine function. Its formula is expressed as:

\[ \csc(x) = \frac{1}{\sin(x)} \]

This trigonometric function is vital in calculus when solving equations that involve derivatives of trigonometric forms. It becomes undefined where the sine function is zero, resulting in vertical asymptotes in the graph.

  • Cosecant is often used in tandem with cotangent in derivatives, especially in the exercise provided, where \( \csc(\frac{3x}{2}) \) heavily influences the first and second derivatives.
  • Understanding its properties helps in dealing with discontinuities and undefined points, essential for full comprehension of concavity and inflection in graphs.

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