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Chapter 7: Problem 54

Identify the correct statements: (A) If \(f(x)=a x^{3}+b\) and \(f\) is strictly increasing on \((-1,1)\) then \(\mathrm{a}>0\). (B) An \(n\) th-degree polynomial has atmost ( \(n-1)\) critical points. (C) If \(\mathrm{f}^{\prime}(\mathrm{x})>0\) for all real numbers \(\mathrm{x}\), then \(\mathrm{f}\) increases without bound. (D) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Short Answer

Expert verified
A) A cubic function in the form \(f(x)=ax^3 + b\) can be strictly increasing on the interval (-1, 1). B) An n-degree polynomial has at most (n-1) critical points. C) If a function has a positive derivative for all real numbers x, the function increases without bound. D) A continuous function with a maximum value in a given interval will have the maximum value at two different points in that interval. Answer: A, B, C

Step by step solution

01

A. Analyzing the cubic function for increasing behavior

To see if the function \(f(x)=ax^3 + b\) is strictly increasing on the interval (-1, 1), we need to consider the first derivative of the function. The first derivative is: \(f'(x)= 3ax^2\). Since \(x^2\) is always non-negative, the sign of the derivative will depend on the sign of \(a\). If \(a>0\), then \(f'(x)>0\) and \(f(x)\) will be strictly increasing on \((-1,1)\). Otherwise, it will be decreasing. Hence, the statement A is correct.
02

B. Polynomial critical points

Critical points of a function occur where the derivative of the function is equal to zero or is undefined. A polynomial function has a derivative that is also a polynomial, specifically of degree \((n-1)\). The maximum number of roots that a polynomial can have, according to the Fundamental Theorem of Algebra, is equal to the degree of the polynomial. Therefore, the derivative of an \(n\)-degree polynomial has at most \((n-1)\) roots, which translates to at most \((n-1)\) critical points. So, Statement B is correct.
03

C. Unbounded increase with positive derivative

Statement C claims that if \(f'(x)>0\) for all real numbers x, then \(f\) increases without bound. Consider the function \(f(x)=e^x\). Its derivative is \(f'(x)=e^x\), which is always positive. And as \(x\to\infty\), \(e^x\to\infty\), which means the function does increase without bound. However, this is just an example; a better way to show this would be to use the Mean Value Theorem. Since \(f'(x)>0\) for all real numbers \(x\), the Mean Value Theorem guarantees that \(f\) is strictly increasing, which means \(f\) increases without bound. Statement C is correct.
04

D. Maximum value occurs at two different points in the interval for a continuous function

For a continuous function on a closed interval, the maximum value can occur at either end of the interval or at a critical point within the interval, due to the Extreme Value Theorem. However, this does not guarantee that a function will have two different maximum values within the same interval. Consider the function \(f(x)=-x^2+4\) on the interval \([-2,2]\). The maximum value is \(4\) and occurs at \(x=0\), which is within the interval. In this case, there is only one maximum value. So, Statement D is incorrect. Based on the evaluation of all the statements, A, B, and C are the correct statements.

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