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Chapter 7: Problem 52

Which of the statements are necessarily true? (A) If \(\mathrm{f}\) is differentiable and \(\mathrm{f}(-1)=\mathrm{f}(1)\), then there is a number \(\mathrm{c}\) such that \(|\mathrm{c}|<\mathrm{l}\) and \(\mathrm{f}^{\prime}(\mathrm{C})=0\). (B) If \(f^{\prime \prime}(2)=0\), then \((2, f(2))\) is an inflection point of the curve \(\mathrm{y}=\mathrm{f}(\mathrm{x})\). (C) There exists a function \(f\) such that \(f(x)>0, f(x)<0\), and $f^{\prime \prime}(x)>0\( for all \)x$. (D) If \(f^{\prime}(x)\) exists and is nonzero for all \(x\), then $f(1) \neq f(0) .$

Short Answer

Expert verified
A) For a differentiable function f, if f(-1) = f(1), there exists a number c such that |c| < 1 and f'(c) = 0. B) If f''(2) = 0, then the point (2, f(2)) is an inflection point. C) There exists a function f such that f(x) > 0, f''(x) < 0, and f(1) = f(0). D) If f'(x) exists and is nonzero for all x, then f(1) 鈮� f(0). Answer: Statements A and D are necessarily true, while statements B and C are not.

Step by step solution

01

Statement A: Rolle's Theorem Application

By Rolle's Theorem, if a function is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there exists a number c in the interval (a, b) such that the derivative f'(c) = 0. In this case, a = -1, b = 1, and f(-1) = f(1). Since f is differentiable, it's also continuous. Therefore, there exists a number c such that |c| < 1 (c is in the interval (-1, 1)) and f'(c) = 0. Statement A is true.
02

Statement B: Inflection Point Analysis

An inflection point is a point on a curve where the concavity of the function changes. If f''(2) = 0, that simply means that the second derivative is 0 at x = 2. This condition by itself is not sufficient to conclude that (2, f(2)) is an inflection point. We also require that f''(x) changes its sign around x = 2. Statement B is not necessarily true.
03

Statement C: Function Properties Examination

Statement C is self-contradictory. A function cannot be simultaneously greater than 0 (f(x) > 0) and less than 0 (f(x) < 0). This statement is false.
04

Statement D: Nonzero Derivative Analysis

If f'(x) exists and is nonzero for all x, that implies that f is strictly increasing or strictly decreasing for all x. This means that f(1) and f(0) cannot be equal, otherwise it would contradict the given condition. Therefore, f(1) 鈮� f(0), and statement D is true. In summary, statements A and D are necessarily true, while statements B and C are not.

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