Chapter 3: Problem 6
Find two positive numbers that satisfy the given requirements. The second number is the reciprocal of the first and the sum is a minimum.
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In Exercises \(101-104,\) use the definition of limits at infinity to prove the limit. $$ \lim _{x \rightarrow-\infty} \frac{1}{x-2}=0 $$
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. $$ s(t)=t^{3}-5 t^{2}+4 t $$
Prove that if \(f\) is differentiable on \((-\infty, \infty)\) and \(f^{\prime}(x)<1\) for all real numbers, then \(f\) has at most one fixed point. A fixed point of a function \(f\) is a real number \(c\) such that \(f(c)=c\).
Consider \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(N\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).
Show that the point of inflection of \(f(x)=x(x-6)^{2}\) lies midway between the relative extrema of \(f\).
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