Problem 12 - A Textbook of Differential Calculus

Question

The total number of points of non-differentiability of \(f(x)=\min \left[|\sin x|,|\cos x|, \frac{1}{4}\right]\) in \((0,2 \pi)\) is (a) 8 (b) 9 (c) 10 (d) 11

Step-by-Step Solution

Answer

Option (c), 10 points of non-differentiability.

Step 1: Identify Points of Non-Differentiability for each Function

Consider the function defined as the minimum of \(|\sin x|\), \(|\cos x|\), and \(\frac{1}{4}\). Each individual function \(|\sin x|\) and \(|\cos x|\) has points where their derivatives do not exist, particularly where the sine and cosine functions are zero. Thus, \(|\sin x|\) is non-differentiable at points where \(\sin x = 0\), which are at \(x = 0, \pi, 2\pi\). Similarly, \(|\cos x|\) is non-differentiable at points where \(\cos x = 0\), which are at \(x = \frac{\pi}{2}, \frac{3\pi}{2}\).

Step 2: Examine Additional Non-Differentiability from Piecewise Definitions

The expression \(\min \left[ |\sin x|, |\cos x|, \frac{1}{4} \right]\) represents a piecewise function. Non-differentiability in piecewise cases occurs at points of intersection or switching. Find intersections: \(|\sin x| = \frac{1}{4}\), \(|\cos x| = \frac{1}{4}\), and \(|\sin x| = |\cos x|\). Check these within \((0, 2\pi)\).

Step 3: Calculate Exact Intersection Points

Calculate \(|\sin x| = \frac{1}{4}\) and \(|\cos x| = \frac{1}{4}\): these occur when \(x = \sin^{-1}\left(\frac{1}{4}\right)\) and \(x = \cos^{-1}\left(\frac{1}{4}\right)\). Numerically, find these points within \((0,2\pi)\). Find intersections where \(|\sin x| = |\cos x|\), which occur at \(x = \frac{\pi}{4}, \frac{5\pi}{4}\).

Step 4: Count Total Points of Non-Differentiability

Consider: \(x = \frac{\pi}{4}, \frac{5\pi}{4}\) where \(|\sin x| = |\cos x|\), along with the intersections of \(|\sin x|\) and \(|\cos x|\) with \(\frac{1}{4}\), as well as previous non-differentiability from steps 1 and 2. Combine these to count all points.

Step 5: Verify Complete Set of Points

Count the unique points: \(x = \sin^{-1}\left(\frac{1}{4}\right), x = \cos^{-1}\left(\frac{1}{4}\right), \frac{\pi}{4}, \frac{5\pi}{4}\), and points \(x = \frac{\pi}{2}, \frac{3\pi}{2}\). Include points such as \(x = \pi\) if they exist between 0 and \(2\pi\). Ensure no duplicates or omissions in the final set.

Key Concepts

Non-Differentiability

In calculus, a function is non-differentiable at a particular point if it lacks a defined tangent there. This means that the function does not have a unique linear approximation at that point. Non-differentiability can occur in several scenarios, such as sharp corners, cusps, or discontinuities in the graph of the function.
The function \(f(x) = \min[|\sin x|, |\cos x|, \frac{1}{4}]\) is a good example, as it involves points of non-differentiability due to absolute values and their intersections.

• Individual non-differentiability: For \(|\sin x|\), non-differentiability is at \(x = 0, \pi, 2\pi\), where the derivative changes abruptly because \(\sin\) transitions from negative to positive and vice versa.
• Similarly, \(|\cos x|\) is non-differentiable at \(x = \frac{\pi}{2}, \frac{3\pi}{2}\).

These points need careful consideration as they define where differentiation, or the slope of the tangent line, becomes problematic.

Piecewise Functions

Piecewise functions are defined by different expressions in different intervals of their domain. They can lead to non-differentiability at the transition points between these expressions. For the function \(f(x) = \min[|\sin x|, |\cos x|, \frac{1}{4}]\), it's vital to identify intersection points where the expressions alternate.

• Non-differentiability occurs at points where \(|\sin x| = \frac{1}{4}\) and \(|\cos x| = \frac{1}{4}\), as these mark changes in the composition of the function within the given intervals.
• The function is also non-differentiable where \(|\sin x| = |\cos x|\), as these points traditionally result in a cusp or corner in piecewise functions.

Using trigonometric identities to find these exact points within a restricted domain \((0, 2\pi)\) is key to understanding the full behavior of the function. By nature, if the function switches from one expression to another, there may be a lack of smoothness or a sharp transition.

Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in differential calculus, exemplified by their periodic behavior and oscillations in the interval \((0, 2\pi)\). These functions provide familiar non-differentiable points due to their wave nature:

• \(\sin x\) reaches zero at multiples of \(\pi\): \(x = 0, \pi, 2\pi\).
• For \(\cos x\), zeros occur at odd multiples of \(\frac{\pi}{2}\): \(x = \frac{\pi}{2}, \frac{3\pi}{2}\).

These classic trigonomic profiles influence the non-differentiability criteria in compound functions like \(|\sin x|\) and \(|\cos x|\), guiding the identification of intervals for piecewise definition. Moreover, trigonometric functions help establish a framework for evaluating limits and continuity, which are crucial in differential calculus.