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Chapter 2: Problem 22

Find the critical points of the function \(f(x)=x-\ell n x+\int_{2}^{x}\left(\frac{1}{z}-2-2 \cos 4 z\right) d z\)

Short Answer

Expert verified
The critical points of the function are given by the following expressions: $$ x = \frac{\cos^{-1}\left(-\frac{1}{2}\right)}{4} + \frac{n\pi}{2} \quad \text{or} \quad x = -\frac{\cos^{-1}\left(-\frac{1}{2}\right)}{4} + \frac{(2n + 1)\pi}{4},$$ where \(n\) is an integer.

Step by step solution

01

Differentiate the function

Differentiate the given function \(f(x)=x-\ell n x+\int_{2}^{x}\left(\frac{1}{z}-2-2 \cos 4 z\right) d z\) with respect to \(x\): $$ f'(x) = \frac{d}{dx} \left( x - \ell n x + \int_{2}^{x} \left( \frac{1}{z} - 2 - 2 \cos 4z \right) dz \right) $$
02

Apply differentiation rules

We apply the sum rule, constant rule, and chain rule to differentiate each term individually: $$ f'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(\ell n x) + \frac{d}{dx} \left( \int_{2}^{x} \left( \frac{1}{z} - 2 - 2 \cos 4z \right) dz \right) $$
03

Differentiate each term

Differentiate \(x\), \(\ell n x\), and the integral term: $$ f'(x) = 1 - \frac{1}{x} + \left( \frac{1}{x} - 2 - 2 \cos 4x \right) $$
04

Simplify \(f'(x)\)

Simplify the expression for the first derivative: $$ f'(x) = 1 - 2 - 2 \cos 4x $$
05

Find the critical points

Set the first derivative equal to 0 to find the critical points: $$ 0 = 1 - 2 - 2 \cos 4x $$
06

Solve for x

Solve for x to find the critical points: $$ 2 \cos 4x = 1 - 2 $$ $$ \cos 4x = -\frac{1}{2} $$ To find the values of \(x\) that satisfy this equation, we can use the inverse cosine function: $$ 4x = \cos^{-1}\left(-\frac{1}{2}\right) + 2n\pi \quad \text{or} \quad 4x = -\cos^{-1}\left(-\frac{1}{2}\right) +(2n+1)\pi $$ Divide both sides by 4: $$ x = \frac{\cos^{-1}\left(-\frac{1}{2}\right)}{4} + \frac{n\pi}{2} \quad \text{or} \quad x = -\frac{\cos^{-1}\left(-\frac{1}{2}\right)}{4} + \frac{(2n + 1)\pi}{4} $$ These expressions give us the critical points of the function.

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Most popular questions from this chapter

It is known that \(\int_{a}^{b} f(x) d x>\int_{a}^{b} g(x) d x\). Does it follow that \(\mathrm{f}(\mathrm{x}) \geq \mathrm{g}(\mathrm{x}) \forall \mathrm{x} \in[\mathrm{a}, \mathrm{b}] ?\) Give examples.

A function \(\mathrm{f}\), continuous on the positive real axis, has the property that \(\int_{1}^{x y} f(t) d t=y \int_{1}^{x} f(t) d t+x \int_{1}^{y} f(t) d t\) for all \(x>0\) and all \(y>0 .\) If \(f(1)=3\), compute \(\mathrm{f}(\mathrm{x})\) for each \(\mathrm{x}>0\).

\(\sqrt{1}+x\) Prove that, if \(\mathrm{n}>1\) (i) \(0<\int_{0}^{\pi / 2} \sin ^{n+1} x d x<\int_{0}^{\pi / 2} \sin ^{n} x d x\), (ii) \(0<\int_{0}^{\pi / 4} \tan ^{n+1} x d x<\int_{0}^{\pi / 4} \tan ^{n} x d x\). (iii) \(0.5<\int_{0}^{1 / 2} \frac{\mathrm{dx}}{\sqrt{\left(1-\mathrm{x}^{2 \mathrm{a}}\right)}}<0.524\).

The linear density of a rod of length \(4 \mathrm{~m}\) is given by \(\rho(\mathrm{x})=9+2 \sqrt{\mathrm{x}}\) measured in kilograms per metre, where \(\mathrm{x}\) is measured in metres from one end of the rod. Find the total mass of the rod.

Prove that, as \(n \rightarrow \infty, \int_{0}^{1} \cos n x \tan ^{-1} x d x \rightarrow 0\).
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