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Chapter 10: Problem 821

If \(\mathrm{f}^{\prime}(\mathrm{x})>0\) and \(\mathrm{g}^{\prime}(\mathrm{x})<0 \mathrm{x} \in \mathrm{R}\) then (a) \(\mathrm{f}(\mathrm{g}(\mathrm{x}))>\mathrm{f}(\mathrm{g}(\mathrm{x}+1))\) (b) \(f(g(x))g(f(x+1))\) (d) \(\mathrm{g}(\mathrm{f}(\mathrm{x}))>\mathrm{g}(\mathrm{f}(\mathrm{x}-1))\)

Short Answer

Expert verified
The short answer based on the given solution is: Statement (a) and statement (c) are true, while statement (b) and statement (d) are false.

Step by step solution

01

Statement (a): \(f(g(x))>f(g(x+1))\)

Take two real numbers, \(x\) and \(x+1\). Since g is a decreasing function (\(g′(x)<0\)), we have \(g(x) > g(x+1)\). Now, since f is an increasing function (\(f'(x)>0\)), if the input increases, the output also increases. Thus, we have \(f(g(x)) > f(g(x+1))\). So, statement (a) is true.
02

Statement (b): \(f(g(x))

This statement is the opposite of statement (a). Since we have proven statement (a) to be true, statement (b) must be false.
03

Statement (c): \(g(f(x))>g(f(x+1))\)

Take two real numbers, \(x\) and \(x+1\). Since f is an increasing function (\(f'(x)>0\)), we have \(f(x) < f(x+1)\). Now, since g is a decreasing function (\(g'(x)<0\)), if the input increases, the output decreases. Thus, we have \(g(f(x)) > g(f(x+1))\). So, statement (c) is true.
04

Statement (d): \(g(f(x))>g(f(x-1))\)

This statement is the opposite of statement (c) with \(x-1\) instead of \(x+1\). Since we have proven statement (c) to be true, statement (d) must be false.

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