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Chapter 2: Problem 20

Let f be a function such that \(f(x)>0\) Assume that f has derivatives of all orders and that \(\ln f(x)=f(x) \int_{0}^{x} f(t) d t\). Find (i) \(\mathrm{f}(0)\), (ii) \(\mathrm{f}^{(1)}(0)\), (iii) \(\mathrm{f}^{2)}(0)\).

Short Answer

Expert verified
Answer: The values for the function \(f(x)\) are given by: (i) \(f(0) = 1\), (ii) \(f^{(1)}(0) = 1\), and (iii) \(f^{(2)}(0) = 0\).

Step by step solution

01

Differentiate both sides of the equation with respect to \(x\)

Apply the chain rule to differentiate the left side of the equation with respect to \(x\): \(\frac{1}{f(x)} \frac{d}{dx}f(x)\) Now, differentiate the right side of the equation using the product rule and the fundamental theorem of calculus: \(\frac{d}{dx}(f(x)\int_{0}^{x} f(t) dt) = f(x) \frac{d}{dx}(\int_{0}^{x} f(t) dt) + \int_{0}^{x} f(t) dt \frac{d}{dx}f(x)\) By the fundamental theorem of calculus, \(\frac{d}{dx}(\int_{0}^{x}f(t)dt)=f(x)\). Thus, the equation becomes: \(\frac{1}{f(x)} \frac{d}{dx}f(x) = f(x) + \int_{0}^{x} f(t) dt \frac{d}{dx}f(x)\) Now we have: (i) \(f(0)\), (ii) \(f^{(1)}(0)\), (iii) \(f^{(2)}(0)\)
02

Find \(f(0)\)

To find \(f(0)\), substitute \(x=0\) into the given equation: \(\ln f(0) = f(0) \int_{0}^{0} f(t) dt\) The integral goes from 0 to 0, which evaluates to 0. Since the natural logarithm of 1 is 0, we have: \(f(0) = 1\)
03

Find \(f^{(1)}(0)\)

Substitute \(x=0\) into the differentiated equation: \(\frac{1}{f(0)} f^{(1)}(0) = f(0) + \int_{0}^{0} f(t) dt f^{(1)}(0)\) Since \(f(0) = 1\), the equation simplifies to: \(f^{(1)}(0) = 1 + 0 \cdot f^{(1)}(0)\) So, \(f^{(1)}(0) = 1\).
04

Differentiate both sides of the equation again with respect to \(x\)

Differentiate both sides of the equation from Step 1: \(\frac{-1}{f^{2}(x)}(f^{(1)}(x))^2 + \frac{1}{f(x)} f^{(2)}(x) = (f^{(1)}(x))^2 + \int_{0}^{x} f(t) dt f^{(2)}(x) + f(x) f^{(1)}(x)\)
05

Find \(f^{(2)}(0)\)

Substitute \(x=0\) into the twice-differentiated equation: \(\frac{-1}{1^2}(1)^2 + \frac{1}{1} f^{(2)}(0) = (1)^2 + \int_{0}^{0} f(t) dt f^{(2)}(0) + 1 \cdot 1\) This equation simplifies to: \(f^{(2)}(0) = 1 - 1 + 0 \cdot f^{(2)}(0)\) So, \(f^{(2)}(0) = 0\). In conclusion, we have: (i) \(f(0) = 1\), (ii) \(f^{(1)}(0) = 1\), and (iii) \(f^{(2)}(0) = 0\).

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Most popular questions from this chapter

Evaluate the following integrals: (i) \(\int_{-1}^{0} \frac{e^{\frac{1}{x}}}{x^{3}} d x\) (ii) \(\int_{-\infty}^{\infty} \frac{1}{e^{x}+e^{-x}} d x\) (iii) \(\int_{3}^{5} \frac{x^{2} d x}{\sqrt{(x-3)(5-x)}}\) (iv) \(\int_{-1}^{1} \frac{d x}{(2-x) \sqrt{1-x^{2}}}\)

\(\int_{-\infty}^{\infty} \mathrm{f}(\mathrm{x}) \mathrm{dx}\) may not equal \(\lim _{\mathrm{b} \rightarrow \infty} \int_{-\mathrm{b}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{d} x\) Show that \(\int_{0}^{\infty} \frac{2 \mathrm{xdx}}{\mathrm{x}^{2}+1}\) diverges and hence that \(\int_{-\infty}^{\infty} \frac{2 x d x}{x^{2}+1}\) diverges. Then show that \(\lim _{b \rightarrow \infty} \int_{-b}^{b} \frac{2 x d x}{x^{2}+1}=0\)

Evaluate the following integrals : (i) \(\int_{1}^{\infty} \frac{d x}{x^{2}(x+1)}\) (ii) \(\int_{0}^{\infty} x^{3} e^{-x^{2}} d x\) (iii) \(\int_{0}^{\frac{1}{6}} \frac{\mathrm{dx}}{\mathrm{x} \ln ^{2} \mathrm{x}}\) (iv) \(\int_{-\infty}^{\infty} \frac{d x}{x^{2}+2 x+2}\)

Given that f satisfies \(|\mathrm{f}(\mathrm{u})-\mathrm{f}(\mathrm{v})| \leq|\mathrm{u}-\mathrm{v}|\) for \(\mathrm{u}\) and \(v\) in \([a, b]\) then prove that (i) \(\mathrm{f}\) is continuous in \([\mathrm{a}, \mathrm{b}]\) and (ii) \(\left|\int_{a}^{b} f(x) d x-(b-a) f(a)\right| \leq \frac{(b-a)^{2}}{2}\).

One of the numbers \(\pi, \pi / 2,35 \pi / 128,1-\pi\) is the correct value of the integral \(\int_{0}^{\pi} \sin ^{8} x d x\). Use the graph of \(\mathrm{y}=\sin ^{8} \mathrm{x}\) and a logical process of elimination to find the correct value.
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