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Chapter 5: Problem 2

Given any constants \(a, b \in \mathbb{R}\) with \(a>b\), find the value of \(x\) at which the difference \(\left(x / \sqrt{x^{2}+a^{2}}\right)-\left(x / \sqrt{x^{2}+b^{2}}\right)\) has the maximum value.

Short Answer

Expert verified
The value of \(x\) at which the difference \(\left(\frac{x}{\sqrt{x^{2}+a^{2}}}\right)-\left(\frac{x}{\sqrt{x^{2}+b^{2}}}\right)\) has the maximum value is \(x = 0\).

Step by step solution

01

Define the given expression and its first derivative

Let us define the given expression as a function of x: $$ f(x) = \left(\frac{x}{\sqrt{x^{2}+a^{2}}}\right)-\left(\frac{x}{\sqrt{x^{2}+b^{2}}}\right) $$ Now we need to find the first derivative of this function, \(f'(x)\). To do so, we can first apply the quotient rule:
02

Apply the quotient rule to differentiate the expression

The quotient rule states that if \(u(x) = \frac{v(x)}{w(x)}\), then \(u'(x) = \frac{v'(x)w(x) - v(x)w'(x)}{(w(x)^2)}\). Applying the quotient rule to differentiate each term of the function \(f(x)\), we obtain: $$ f'(x) = \frac{\sqrt{x^2 + a^2} - \frac{x^2}{\sqrt{x^2 + a^2}} - \sqrt{x^2 + b^2} + \frac{x^2}{\sqrt{x^2 + b^2}}}{\left(\sqrt{x^2 + a^2}\right)\left(\sqrt{x^2 + b^2}\right)} $$
03

Simplify the first derivative

To simplify the first derivative, we can combine the terms: $$ f'(x) = \frac{x^2 (a^2 - b^2)}{\sqrt{x^2 + a^2}\left(\sqrt{x^2 + b^2}\right)(\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2})} $$
04

Find the critical points

To find the critical points, we set the first derivative equal to zero and solve for x: $$ 0 = \frac{x^2 (a^2 - b^2)}{\sqrt{x^2 + a^2}\left(\sqrt{x^2 + b^2}\right)(\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2})} $$ Since \(a \neq b\), we can divide both sides by \((a^2 - b^2)\): $$ 0 = \frac{x^2}{\sqrt{x^2 + a^2}\left(\sqrt{x^2 + b^2}\right)(\sqrt{x^2 + a^2} + \sqrt{x^2 + b^2})} $$ The critical point occurs when \(x = 0\).
05

Determine the maximum value

To find if the critical point corresponds to a maximum value of the difference expression, we can use the second derivative test. However, we can also notice that \(f'(x)\) is symmetric around x = 0, meaning that the difference expression is increasing for \(x < 0\) and decreasing for \(x > 0\), thus confirming that \(x = 0\) corresponds to the maximum value of the difference. So, the value of the \(x\) at which the difference \(\left(\frac{x}{\sqrt{x^{2}+a^{2}}}\right)-\left(\frac{x}{\sqrt{x^{2}+b^{2}}}\right)\) has the maximum value is \(x = 0\).

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Most popular questions from this chapter

Consider \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=x^{2}-2 x-3 .\) If \(x_{0} \neq 1\), then show that the Newton sequence with its initial point \(x_{0}\) converges to \(-1\) if \(x_{0}<1\), and to 3 if \(x_{0}>1\),

Let \(I\) be an interval in \(\mathbb{R}, f: I \rightarrow \mathbb{R}\), and \(c\) be an interior point of \(I\). (i) Suppose there is \(n \in \mathbb{N}\) such that \(f^{(2 n)}\) exists at \(c\), and \(f^{\prime}(c)=f^{\prime \prime}(c)=\) \(\cdots=f^{(2 n-1)}(c)=0 .\) If \(f^{(2 n)}(c)<0\), then show that \(f\) has a strict local maximum at \(c\), whereas if \(f^{(2 n)}(c)>0\), then show that \(f\) has a strict local minimum at \(c\). (Hint: Taylor's formula.) (ii) Suppose there is \(n \in \mathbb{N}\) such that \(f^{(2 n+1)}\) exists at \(c\), and \(f^{\prime \prime}(c)=\) \(f^{\prime \prime \prime}(c)=\cdots=f^{(2 n)}(c)=0\). If \(f^{(2 n+1)}(c) \neq 0\), then show that \(c\) is a strict point of inflection for \(f\). (Hint: Taylor's formula.) (iii) Suppose that \(f\) is infinitely differentiable at \(c\) and \(f^{\prime}(c)=0\), but \(f^{(k)}(c) \neq 0\) for some \(k \in \mathbb{N} .\) Show that either \(f\) has a strict local extremum at \(c\), or \(c\) is a strict point of inflection for \(f\).

Let \(f:[a, b] \rightarrow[a, b]\) be continuous and monotonic. Then show that for any \(x_{0} \in[a, b]\), the Picard sequence for \(f\) with its initial point \(x_{0}\) converges to a fixed point of \(f\). (Hint: Show that the Picard sequence \(\left(x_{n}\right)\) is monotonic by considering separately the cases \(x_{0} \leq x_{1}\) and \(x_{0} \geq x_{1}\).)

Consider the following functions: (i) \(f(x):=\sqrt{1+x}, \quad x \geq-1\) (ii) \(f(x):=1 / \sqrt{1-x}, \quad x \leq 1\). For each of them, find: (a) The linear approximation \(L(x)\) around 0 . (b) An estimate for the error \(e_{1}(x)\) when \(x>0\) and when \(x<0\). Also, find an upper bound for \(\left|e_{1}(x)\right|\) that is valid for all \(x \in(0,0.1)\), and an upper bound for \(\left|e_{1}(x)\right|\) that is valid for all \(x \in(-0.1,0)\). (c) The quadratic approximation \(Q(x)\) around 0 . (d) An estimate for the error \(e_{2}(x)\) when \(x>0\) and when \(x<0\). Also, an upper bound for \(\left|e_{2}(x)\right|\) that is valid for all \(x \in(0,0.1)\), and an upper bound for \(\left|e_{2}(x)\right|\) that is valid for all \(x \in(-0.1,0)\).

Let \(\left(x_{n}\right)\) be a sequence in \(\mathbb{R}\) and \(c \in \mathbb{R}\) such that \(x_{n} \rightarrow c\). Assume that there is \(n_{0} \in \mathbb{N}\) such that \(x_{n} \neq c\) for all \(n \geq n_{0}\). If there is a real number \(p\) such that $$ \alpha:=\lim _{n \rightarrow \infty} \frac{\left|x_{n}-c\right|}{\left|x_{n-1}-c\right|^{p}} $$ exists and is nonzero, then \(p\) is called the order of convergence of \(\left(x_{n}\right)\) to \(c\) and \(\alpha\) is called the corresponding asymptotic error constant. (i) Let \(f:(a, b) \rightarrow(a, b)\) and \(x_{0} \in(a, b)\) be such that the Picard sequence \(\left(x_{n}\right)\) for \(f\) with its initial point \(x_{0}\) converges to some \(x^{*} \in(a, b)\). If \(f\) is continuous at \(x^{*}\), then show that \(x^{*}\) is a fixed point of \(f\). Further, if if is \(p\) times differentiable at \(x^{*}\) and \(f^{\prime}\left(x^{*}\right)=\cdots=f^{(p-1)}\left(x^{*}\right)=0\) but \(f^{(p)}\left(x^{*}\right) \neq 0\), then show that the order of convergence of \(\left(x_{n}\right)\) to \(x^{*}\) is \(p\) and the corresponding asymptotic error constant is \(\left|f^{(p)}\left(x^{*}\right)\right| / p !\) (Hint: Note that \(x_{n}-x^{*}=f\left(x_{n-1}\right)-f\left(x^{*}\right)-f^{\prime}\left(x^{*}\right)\left(x_{n-1}-x^{*}\right)-\) \(\cdots-f^{(p-1)}\left(x^{*}\right)\left(x_{n-1}-x^{*}\right)^{p-1} /(p-1) !\), and use L'H么pital's Rule.) (ii) Let \(f:(a, b) \rightarrow \mathbb{R}\) and \(x_{0} \in(a, b)\) be such that \(f\) is differentiable and \(f^{\prime}(x) \neq 0\) for all \(x \in(a, b)\). Assume that the Newton sequence \(\left(x_{n}\right)\) for \(f\) with its initial point \(x_{0}\) converges to some \(r \in(a, b)\). If \(f^{\prime}\) is bounded on \((a, b)\), then show that \(r\) is a solution of \(f(x)=0\). Further, if \(f\) is twice differentiable at \(r\) and \(f^{\prime \prime}(r) \neq 0\), then show that the order of convergence of \(\left(x_{n}\right)\) to \(r\) is 2, and the corresponding asymptotic error constant is \(\left|f^{\prime \prime}(r)\right| / 2\left|f^{\prime}(r)\right| .\) (Hint: Note that \(f\left(x_{n-1}\right)\left(x_{n}-r\right)=\) \(\left[f^{\prime}\left(x_{n-1}\right)-\left(f\left(x_{n-1}\right)-f(r)\right) /\left(x_{n-1}-r\right)\right]\left(x_{n-1}-r\right)\) for \(\left.n \in \mathbb{N} .\right)\)
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