Question

\(\mathrm{f}(\mathrm{x})=|[\mathrm{x}] \mathrm{x}|,-1 \leq \mathrm{x} \leq 2\) then (a) continuous at \(\mathrm{x}=0\) (b) discontinuous at \(\mathrm{x}=0\) (c) diffemtiable at \(\mathrm{x}=0\) (d) continuous at \(\mathrm{x}=2\)

Short answer

The function \(f(x)=|[x]x|\) is continuous at \(x=0\) and \(x=2\) (options (a) and (d)), but it is not differentiable at \(x=0\) (not option (c)).

Step-by-step solution

Definition of the given function

The function is given as \(\mathrm{f}(\mathrm{x})=|[\mathrm{x}] \mathrm{x}|\). Here, [x] denotes the greatest integer less than or equal to x.

Explore the behavior of the function in the given intervals

Given the range -1 ≤ x ≤ 2, let's break down the behavior of the function in different intervals: 1. \(-1 \leq x < 0\): In this interval, [x] = -1, and the function becomes \(\mathrm{f}(\mathrm{x})=|(-1) \mathrm{x}| = |-x|\) 2. \(0 \leq x < 1\): In this interval, [x] = 0, and the function becomes \(\mathrm{f}(\mathrm{x})=|(0) \mathrm{x}| = 0\) 3. \(1 \leq x \leq 2\): In this interval, [x] = 1, and the function becomes \(\mathrm{f}(\mathrm{x})=|(1) \mathrm{x}| = |x|\) Now that we have the function behavior in each interval, we can check for continuity and differentiability at x=0 and x=2.

Determine continuity at x=0 and x=2

For x=0: We have the left-hand limit as x approaches 0: \(\lim_{x \to 0^-} |-x| = 0\) And the right-hand limit as x approaches 0: \(\lim_{x \to 0^+} 0 = 0\) Since both left and right limits are 0 and the function is defined at x=0 as f(0)=0, the function is continuous at x=0, and the correct answer is (a). For x=2: We have the left-hand limit as x approaches 2: \(\lim_{x \to 2^-} |x| = 2\) The function is defined at x=2 as f(2)=2, so the function is continuous at x=2, and the correct answer is (d).

Determine differentiability at x=0

To determine if the function is differentiable at x=0, we need to check if the derivative exists at this point. Using the function behavior determined in Step 2: 1. For \(-1 \leq x < 0\), the function is f(x) = |-x|, so f'(x) = -1. 2. For \(0 \leq x < 1\), the function is f(x) = 0, so f'(x) = 0. Let's check the left-hand and right-hand derivatives at x=0: The left-hand derivative at x=0: \(f'(0^-) = -1\) The right-hand derivative at x=0: \(f'(0^+) = 0\) Since the left-hand derivative and the right-hand derivative at x=0 are not equal, the function is not differentiable at x=0, and the correct answer is not (c). So, the final answers are (a) and (d). The function is continuous at x=0 and x=2, but not differentiable at x=0.

Key concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is a simple yet vital function to comprehend. When you apply this function to a number \(x\), it returns the largest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([-2.3] = -3\). The greatest integer function is denoted by square brackets, like this: \[ [x] \].

One of the key characteristics of the greatest integer function is that it creates a step-like graph, where each step is an integer value. This gives rise to discontinuities at each integer point where the function steps from one integer value to the next.

In the context of the given exercise, the greatest integer function is combined with the variable \(x\), forming expressions such as |[x]x|. Understanding how the greatest integer function works helps us determine its influence on the function’s continuity and differentiability.

Piecewise Function

A piecewise function is a function that has different expressions based on the input or the interval. This type of function allows us to define distinct behaviors over specified ranges of input values.

In the given exercise, we break down \[f(x)=|[x]x|\] into three separate expressions depending on the interval:

• For \(-1 \leq x < 0\), \[f(x) = |-x|\]
• For \(0 \leq x < 1\), \[f(x) = 0\]
• For \(1 \leq x \leq 2\), \[f(x) = |x|\]

Piecewise functions help us analyze the properties like continuity and differentiability by simplifying complex functions into manageable parts. By understanding each piece, you can determine how a function behaves in each interval and at the boundaries between the pieces.

Differentiability

Differentiability is the property of a function that allows it to be differentiated, which means finding the derivative of that function. For a function to be differentiable at a given point, the derivative must exist at that point. This involves checking if the left-hand and right-hand derivatives are equal at that point.

In the problem at hand, we examined the differentiability of \[f(x)=|[x]x|\] at \(x=0\).

Here’s what we found:

• For \(-1 \leq x < 0\), the derivative \[f'(x) = -1\]
• For \(0 \leq x < 1\), the derivative \[f'(x) = 0\]

Checking the left-hand and right-hand derivatives at \(x=0\), we find: \[f'(0^-) = -1\] and \[f'(0^+) = 0\].

Since the derivatives are not equal, the function is not differentiable at \(x=0\). Differentiability is an essential concept in calculus, enabling us to analyze and understand the behavior of functions more deeply.

Limit Analysis

Limit analysis is a technique used to study the behavior of a function as it approaches a specific input value. Limits help us understand continuity and differentiability by examining how function values behave as they near a particular point.

In this exercise, we used limit analysis to check continuity at \(x=0\) and \(x=2\):

To confirm continuity at \(x=0\):

• Calculate the left-hand limit: \(\lim_{x \to 0^-} |-x| = 0\)
• Calculate the right-hand limit: \(\lim_{x \to 0^+} 0 = 0\)

Both limits agree, and the function equals these limits at \(x=0\), thus, the function is continuous there.

At \(x=2\), the left-hand limit of \(|x|\) is 2, and since the function value \(f(2)=2\), it is continuous at \(x=2\) as well.

Limit analysis is especially crucial in calculus to discuss function behavior near points of interest, helping ascertain properties like continuity and differentiability.