Problem 16 - Essential Calculus

Question

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=2 \csc \frac{3 x}{2}, \quad(0,2 \pi)\)

Step-by-Step Solution

Answer

After taking the second derivative, setting it equal to zero, and finding discontinuities, the function's concavity changes at the inflection points. These points, as well as the intervals of concavity, are found by testing the second derivative on both sides of these points.

Step 1: Differentiate the function.

To find the points of inflection, we first have to find the second derivative of the function. But before that, let's find the first derivative. \The derivative of csc(u), in general, is \(-\csc(u) \cot(u) \cdot u'\), where \(u'\) is the derivative of \(u\). \For \(f(x) = 2 \csc( \frac{3x}{2})\), let \(u = \frac{3x}{2}\) and \(u' = \frac{3}{2}\). So, \ \(f'(x) = -2\[\csc(\frac{3x}{2}) \cot(\frac{3x}{2})\] * \frac{3}{2}\

Step 2: Compute the second derivative.

Again using trigonometry rules, since the derivative of \(\csc(u)\) is \(-\csc(u)\cot(u)\) and of \(\cot(u)\) is \(-\csc^2(u)\), the second derivative will be:\(f''(x) = \[\frac{3}{2} \csc(\frac{3x}{2}) \cot(\frac{3x}{2})\][2 \csc(\frac{3x}{2}) \cot(\frac{3x}{2}) - \frac{3}{2} \csc^2(\frac{3x}{2})]\)

Step 3: Find the points where the second derivative equals 0 find undefined points

This is done by solving the equation \(f''(x) = 0\) and find points where the derivative is undefined. For the function \(f''(x)\) to equal zero, the function inside the brackets has to equal zero, because \(\csc(x)\) is never zero. So, place \(2 \csc(\frac{3x}{2}) \cot(\frac{3x}{2}) - \frac{3}{2} \csc^2(\frac{3x}{2}) = 0\) and solve for \(x\).

Step 4: Analyze the concavity and inflection points

Use the results from Step 3 to find the points of inflection, where the graph changes concavity, and the intervals where the graph is concave up or concave down. By substituting the values of \(x\) found in step 3 into \(f''(x)\) and analyzing whether \(f''(x)\) has different signs on either side of the point. Create a sign chart for the concavity and note that if it changes from + to - , the graph is concave down and if it changes from - to +, the function is concave up.