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Chapter 1: Problem 25

Consider \(D \subseteq \mathbb{R}\) and \(f: D \rightarrow \mathbb{R}\) defined by the following. Determine whether \(f\) is bounded above on \(D .\) If yes, find an upper bound for \(f\) on \(D\). Also, determine whether \(f\) is bounded below on \(D .\) If yes, find a lower bound for \(f\) on \(D\). Also, determine whether \(f\) attains its upper bound or lower bound. (i) \(D=(-1,1)\) and \(f(x)=x^{2}-1\), (ii) \(D=(-1,1)\) and \(f(x)=x^{3}-1\), (iii) \(D=(-1,1]\) and \(f(x)=x^{2}-2 x-3\), (iv) \(D=\mathbb{R}\) and \(f(x)=\frac{1}{1+x^{2}}\).

Short Answer

Expert verified
(i) For the function \(f(x) = x^2 - 1\) on \((-1, 1)\), it is bounded above by -1, attained at \(x = 0\). The function has no lower bound in the given interval. (ii) For the function \(f(x) = x^3 - 1\) on \((-1, 1)\), it has no upper bound. The lower bound is -2, attained at \(x = -1\). (iii) For the function \(f(x) = x^2 - 2x - 3\) on \((-1, 1]\), it is bounded above by -4, attained at \(x = 1\). The function has no lower bound in the given interval. (iv) For the function \(f(x) = \frac{1}{1 + x^2}\) on \(\mathbb{R}\), it is bounded above by 1, attained at \(x = 0\). The lower bound is 0, which is asymptotically approached but never reached.

Step by step solution

01

(i) Analyzing \(f(x) = x^2 - 1\)

For \(x \in (-1, 1)\), note that the function is a parabola opening upwards and its vertex is at \(x = 0\). Let's find the value of the function at this point.
02

(ii) Analyzing \(f(x) = x^3 - 1\)

For \(x \in (-1, 1)\), the function is a cubic function with one real root at \(x = 1\). It is monotonically increasing over the given interval.
03

(iii) Analyzing \(f(x) = x^2 - 2x - 3\)

For \(x \in (-1, 1]\), note that the function is a parabola opening upwards with vertex at \(x = 1\). As \(1\) is also included in the domain, we can also evaluate the function at its vertex.
04

(iv) Analyzing \(f(x) = \frac{1}{1 + x^2}\)

For \(x \in \mathbb{R}\), this is a rational function with a horizontal asymptote at \(y = 0\). The function is symmetric with respect to the y-axis and has maximum value at \(x = 0\). #Step 2 - Boundedness and Finding Bounds#
05

(i) Bounds for \(f(x) = x^2 - 1\)

On \(-1 < x < 1\), the function has a minimum value at \(x = 0\) and is symmetric with respect to the y-axis. Therefore, it is bounded above by its value at the center: Upper bound: \(f(0) = 0^2 - 1 = -1\). As for its lower bound, since the function is a parabola opening upwards, it does not have a lower bound in the given interval.
06

(ii) Bounds for \(f(x) = x^3 - 1\)

As the function is monotonically increasing in the given interval, we can check its maximum value at \(x = 1\) (not included) and minimum value at \(x = -1\): Upper bound: As the interval is open, the function does not have an upper bound in the given interval. Lower bound: \(f(-1) = (-1)^3 - 1 = -2\).
07

(iii) Bounds for \(f(x) = x^2 - 2x - 3\)

The function has a minimum value at the vertex, which is \(x = 1\), and the interval is closed at \(x = 1\). Therefore, the function is bounded both above and below: Upper bound: \(f(1) = 1^2 - 2(1) - 3 = -4\). Lower bound: As the function is a parabola opening upwards, it does not have a lower bound in the given interval.
08

(iv) Bounds for \(f(x) = \frac{1}{1 + x^2}\)

The function is symmetric and has a maximum value at \(x = 0\). Also, it has a horizontal asymptote of \(y = 0\): Upper bound: \(f(0) = \frac{1}{1 + 0^2} = 1\). Lower bound: \(0\). #Step 3 - Attainment of Bounds#
09

(i) Attainment of Bounds for \(f(x) = x^2 - 1\)

Upper bound: Yes, the function attains its upper bound at \(x = 0\). Lower bound: No, there is no lower bound in the given interval.
10

(ii) Attainment of Bounds for \(f(x) = x^3 - 1\)

Upper bound: No, there is no upper bound in the given interval. Lower bound: Yes, the function attains its lower bound at \(x = -1\).
11

(iii) Attainment of Bounds for \(f(x) = x^2 - 2x - 3\)

Upper bound: Yes, the function attains its upper bound at \(x = 1\). Lower bound: No, there is no lower bound in the given interval.
12

(iv) Attainment of Bounds for \(f(x) = \frac{1}{1 + x^2}\)

Upper bound: Yes, the function attains its upper bound at \(x = 0\). Lower bound: No, the function asymptotically approaches the lower bound, but never reaches it.

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Most popular questions from this chapter

Let \(I\) be an interval and \(f: I \rightarrow \mathbb{R}\) be a monotonically increasing function. Given any \(r \in \mathbb{R}\), show that \(r f: I \rightarrow \mathbb{R}\) is a monotonically increasing function if \(r \geq 0\) and a monotonically decreasing function if \(r<0\).

If \(m, n, n^{\prime} \in \mathbb{Z}\) are such that \(m\) and \(n\) are relatively prime and \(m \mid n n^{\prime}\), then show that \(m \mid n^{\prime}\). Deduce that if \(p\) is a prime (which means that \(p\) is an integer \(>1\) and the only positive integers that divide \(p\) are 1 and \(p\) ) and if \(p\) divides a product of two integers, then it divides one of them. (Hint: Exercise 38.)

Consider \(D \subseteq \mathbb{R}\) and \(f: D \rightarrow \mathbb{R}\) defined by the following. Determine whether \(f\) is bounded above on \(D .\) If yes, find an upper bound for \(f\) on \(D\). Also, determine whether \(f\) is bounded below on \(D .\) If yes, find a lower bound for \(f\) on \(D\). Also, determine whether \(f\) attains its upper bound or lower bound. (i) \(D=(-1,1)\) and \(f(x)=x^{2}-1\), (ii) \(D=(-1,1)\) and \(f(x)=x^{3}-1\), (iii) \(D=(-1,1]\) and \(f(x)=x^{2}-2 x-3\), (iv) \(D=\mathbb{R}\) and \(f(x)=\frac{1}{1+x^{2}}\).

Given any function \(f: \mathbb{R} \rightarrow \mathbb{R}\), prove the following: (i) If \(f(x y)=f(x)+f(y)\) for all \(x, y \in \mathbb{R}\), then \(f(x)=0\) for all \(x \in \mathbb{R}\). [Note: It is, however, possible that there are nonzero functions defined on subsets of \(\mathbb{R}\), such as \((0, \infty)\) that satisfy \(f(x y)=f(x)+f(y)\) for all \(x, y\) in the domain of \(f\). A prominent example of this is the logarithmic function, which will be discussed in Section 7.1.] (ii) If \(f(x y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\), then either \(f(x)=0\) for all \(x \in \mathbb{R}\), or \(f(x)=1\) for all \(x \in \mathbb{R}\), or \(f(0)=0\) and \(f(1)=1\). Further, \(f\) is either an even function or an odd function. Give examples of even as well as odd functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) satisfying \(f(x y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\)

Let \(f:(0, \infty) \rightarrow \mathbb{R}\) be the function defined by the following: (i) \(f(x)=\sqrt{x}\), (ii) \(f(x)=x^{3 / 2}\), (iii) \(f(x)=\frac{1}{|x|}\), (iv) \(f(x)=\frac{1}{x^{2}}\). Sketch the graph of \(f\) and determine the points at which \(f\) has local extrema as well as the points of inflection of \(f\), if any, in each case.
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