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Chapter 5: Problem 54

Let \(f(x)=\left\\{\begin{array}{cl}\frac{(x-1)(6 x-1)}{2 x-1} & \text { if } x \neq \frac{1}{2} \\ 0 & \text { if } x=\frac{1}{2}\end{array}\right.\), then at \(x=\frac{1}{2}\) (a) \(f\) has a local maxima (b) \(f\) has a local minima (c) \(f\) has an inflection point (d) \(f\) has a removable discontinuity

Short Answer

Expert verified
The short answer to the question is: (d) \(f(x)\) has a removable discontinuity at \(x=\frac{1}{2}\)

Step by step solution

01

Evaluate the left-hand limit of f(x) at x = 1/2

At \(x=\frac{1}{2}\), the function takes the form \(\frac{(x-1)(6 x-1)}{2 x-1}\) when \(x\neq\frac{1}{2}\). First, evaluate the left-hand limit as x gets closer to \(\frac{1}{2}\): \[ \lim _{x \rightarrow \frac{1}{2}^{-}} f(x)=\lim _{x \rightarrow \frac{1}{2}^{-}} \frac{(x-1)(6 x-1)}{2 x-1} \]
02

Evaluate the right-hand limit of f(x) at x = 1/2

Next, evaluate the right-hand limit as x gets closer to \(\frac{1}{2}\): \[ \lim _{x \rightarrow \frac{1}{2}^{+}} f(x)=\lim _{x \rightarrow \frac{1}{2}^{+}} \frac{(x-1)(6 x-1)}{2 x-1} \]
03

Determine if the left-hand and right-hand limits are equal

Compare the values obtained in Steps 1 and 2, if they are equal then the limit exists at \(x=\frac{1}{2}\), else it doesn't: \[ \lim _{x \rightarrow \frac{1}{2}^{-}} f(x) = \lim _{x \rightarrow \frac{1}{2}^{+}} f(x) \]
04

Determine the nature of the discontinuity, if any and answer the question

If the limits in Step 3 are equal, evaluate the nature of the discontinuity by comparing the value of the function at \(x=\frac{1}{2}\) (i.e., \(f(\frac{1}{2})=0\)) with the obtained limits. Else, there is no discontinuity and we can look for a possible local maxima or minima.

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Key Concepts

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

Limits of a Function
Understanding the concept of limits is crucial in calculus. A limit describes the value that a function (or sequence) approaches as the input (or index) approaches some value. In formal terms, we say the limit of a function f(x) as x approaches a is L, if we can make f(x) arbitrarily close to L by taking x sufficiently close to a. This concept is symbolically denoted as
\[\begin{equation}\lim_{{x \to a}} f(x) = L\end{equation}\]
In practical terms, this means that when you zoom in on the graph of the function at x = a, the values of f(x) get closer and closer to L, even if f(a) itself is different from L or even undefined.
Left-Hand Limit
The left-hand limit of a function at a certain point is the value that the function approaches as the input approaches that point from the left side. Mathematically, we represent the left-hand limit of f(x) as x approaches a from the left (a-) as:
\[\begin{equation}\lim_{{x \to a^{-}}} f(x)\end{equation}\]
It is important to understand that for the limit to exist at the point a, the left-hand limit and the right-hand limit must be equal, meaning they should approach the same value from both sides of a.
Right-Hand Limit
In contrast to the left-hand limit, the right-hand limit refers to the value that f(x) is approaching as x gets closer to a from the right. For such a limit, we formulate:
\[\begin{equation}\lim_{{x \to a^{+}}} f(x)\end{equation}\]
This concept is crucial because sometimes a function can behave differently on each side of a point. For the overall limit at a point to exist, the right-hand limit must match the left-hand limit exactly.
Local Maxima and Minima
When studying functions, we often encounter peaks and valleys in their graphs, which are called local maxima and local minima, respectively. A local maximum is a point where the function takes a greater value than at any nearby points, while a local minimum is one where the function takes a smaller value. These points are important in differential calculus because they represent where a function changes its direction. To find them, we look for points where the derivative of the function is zero (indicating a flat slope) and apply various tests to determine the nature of those points.
Discontinuities in Functions
A discontinuity occurs when a function abruptly changes its value, has a break, or has a 'hole' at a certain point. There are several types of discontinuities, but one that is particularly significant in calculus is the removable discontinuity. This type happens when the limit of the function exists at a point (both sides agree), but the value of the function at the very point is different or undefined. In essence, a removable discontinuity can be 'fixed' by redefining the function at that point to be equal to the limit. Understanding these breaks in the function helps us to better understand the overall behavior and properties of the function.
Differential Calculus
At the heart of understanding change, differential calculus is a major branch of mathematics that deals with rates of change and slopes of curves. When applied to a function, it involves calculating derivatives, which are measures of how much a function changes as its input changes. Through this process, we can find slope values at specific points on a graph, allowing us to identify and analyze local maxima and minima, motion under changing forces, and many other physical and theoretical situations. Differential calculus is fundamental to many scientific fields, including physics, engineering, economics, and beyond, providing a toolset for modeling and solving problems involving change.

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Most popular questions from this chapter

A point \(P\) is given on the circumference of a circle of radius \(r\). Chords \(Q R\) are parallel to the tangent at \(\mathrm{P}\). Determine the maximum possible area of the triangle \(P Q R\)

A right triangle is drawn in a semicircle of radius \(\frac{1}{2}\) with one of its legs along the diameter. The maximum area of the triangle is (a) \(\frac{1}{4}\) (b) \(\frac{3 \sqrt{3}}{32}\) (c) \(\frac{3 \sqrt{3}}{16}\) (d) \(\frac{1}{8}\)

The scmi-vertical angle of a right cone of given total surface (including area of base) and max. volume is (a) \(\sin ^{-1} \frac{1}{3}\) (b) \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\) (c) \(45^{\circ}\) (d) \(30^{\circ}\)

A window of fixed perimeter (inducing the base of the arc) is in the form of a rectangle surmounted by a semi-circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio of the sides of the rectangle so that the window transmits the maximum light?

A: If \(f(x)=\max \left\\{x^{2}-2 x+2,|x-1|\right\\}\) the greatest value of \(f(x)\) on the interval \([0,3]\) is 5 R: Greatest value \(=f(3)=\max \\{5,2\\}=5\).
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