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Chapter 10: Problem 885

If \(\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|\) any \(\mathrm{g}(\mathrm{x})=\mathrm{f}\left(\mathrm{f}(\mathrm{f}(\mathrm{x}))\right.\) the for \(\mathrm{x}>2, \mathrm{~g}^{\prime}(\mathrm{x})\) is equal to (a) 1 for all \(x>2\) (b) 1 for \(2

Short Answer

Expert verified
The derivative of g(x) for x > 2 is 1 for all x > 2, which corresponds to option (a).

Step by step solution

01

Express g(x) in terms of f(x)

Let's first write the function g(x) in terms of f(x). \( g(x) = f(f(f(x))) \)
02

Find f(x)

Now, let's find the expression for f(x). \( f(x) = |x - 1| \) Since we only need to consider x>2, we can simplify the expression for f(x): \( f(x) = x - 1 \), where x>2.
03

Find f(f(x))

Let's substitute f(x) into itself to find f(f(x)). \( f(f(x)) = f(x-1) = (x-1) - 1 = x - 2 \)
04

Find g(x) = f(f(f(x)))

Now, let's substitute f(f(x)) into f(x) to find g(x). \( g(x) = f(x-2) = (x-2) - 1 = x - 3 \) Now we have the function g(x) = x - 3.
05

Find the derivative of g(x)

Now we will find the derivative of g(x) with respect to x, that is, g'(x). \( g'(x) = \frac{d}{dx}(x - 3) = 1 \) So, we got g'(x) = 1. As we were working for x > 2, our answer is: (a) 1 for all x > 2.

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Most popular questions from this chapter

If \(\mathrm{f}(\mathrm{x})=[\mathrm{x} /(\sin \mathrm{x})]\) and \(\mathrm{g}(\mathrm{x})=[\mathrm{x} /(\tan \mathrm{x})]\) where \(0<\mathrm{x} \leq 1\) than in this interval (a) both \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are increasing function (b) both \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are decreasing function (c) \(\mathrm{f}(\mathrm{x})\) is an increasing function (d) \(\mathrm{g}(\mathrm{x})\) is an increasing function

In \(\mathrm{a}+\mathrm{b}+\mathrm{c}=0\), than the equation \(3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) has, in the interval \((0,1)\) (a) at least one root (b) at most one root (c) no root (d) Exactly one root exist.

If \(f\) is an even function and \(f^{\prime}(x)\) is define than \(f^{\prime}(x)+f^{\prime}(-x)\) (a) 0 (b) \(<0\) \((\mathrm{c}) \neq 0\) \((\mathrm{d})>0\)

If \(\mathrm{y}=\mathrm{x} \tan (\mathrm{x} / 2)\) and \(\mathrm{A}(\mathrm{dy} / \mathrm{dx})-\mathrm{B}=\mathrm{x}\) then \((\mathrm{B} / \mathrm{A})=\) (a) \(\cot (\mathrm{x} / 2)\) (b) \(\tan (\mathrm{x} / 2)\) (c) \(\tan \mathrm{x}\) (d) \(\cot x\)

\((\mathrm{d} / \mathrm{d} \mathrm{x})\left[(3 / 4) \cos \mathrm{x}-\cos ^{3} \mathrm{x}\right]\) when \(\mathrm{x}=18^{\circ}\) and \(\sin 54=\sqrt{(5+1)}\) (a) \((3 / 4) \sqrt{5}\) (b) \(3(\sqrt{5}+1)\) (c) \((3 / 16)(\sqrt{5}+1)\) (d) \((3 / 4)(\sqrt{5}+1)\)
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