91Ó°ÊÓ

Show that the point of inflection of \(f(x)=x(x-6)^{2}\) lies midway between the relative extrema of \(f\).

Prove that every cubic function with three distinct real zeros has a point of inflection whose \(x\) -coordinate is the average of the three zeros.

Physics Newton's First Law of Motion and Einstein's Special Theory of Relativity differ concerning a particle's behavior as its velocity approaches the speed of light, \(c\). Functions \(N\) and \(E\) represent the predicted velocity, \(v\), with respect to time, \(t\), for a particle accelerated by a constant force. Write a limit statement that describes each theory.

Motion Along a Line, the function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$s(t)=t^{3}-20 t^{2}+128 t-280$$

A line with slope \(m\) passes through the point \((0,-2)\). (a) Write the distance \(d\) between the line and the point \((4,2)\) as a function of \(m\). (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow-\infty} d(m) .\) Interpret the results geometrically.

Creating Polynomial Functions, find a polynomial function $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} $$ that has only the specified extrema. (a) Determine the minimum degree of the function and give the criteria you used in determining the degree. (b) Using the fact that the coordinates of the extrema are solution points of the function, and that the \(x\) -coordinates are critical numbers, determine a system of linear equations whose solution yields the coefficients of the required function. (c) Use a graphing utility to solve the system of equations and determine the function. (d) Use a graphing utility to confirm your result graphically. Relative minimum: \((1,2)\) Relative maxima: \((-1,4),(3,4)\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. An \(n\) th-degree polynomial has at most \((n-1)\) critical numbers.

Use the definition of limits at infinity to prove the limit. $$ \begin{aligned} &\text { Prove that if } p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0} \text { and }\\\ &q(x)=b_{m} x^{m}+\cdots+b_{1} x+b_{0}\left(a_{n} \neq 0, b_{m} \neq 0\right), \text { then }\\\ &\lim _{x \rightarrow \infty} \frac{p(x)}{q(x)}=\left\\{\begin{array}{ll} 0, & nm \end{array}\right. \end{aligned} $$