91Ó°ÊÓ

Find all critical numbers of the given function. $$ f(x)=x^{2}+4 x+6 $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=-\frac{3}{2} x^{2}+x $$

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(x)=x^{2}+6 x-11 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} 2 /(x-3) $$

In an autocatalytic chemical reaction a substance \(A\) is converted into a substance \(B\) in such a manner that $$ \frac{d x}{d t}=k x(a-x) $$ where \(x\) is the concentration of substance \(B\) at time \(t, a\) is the initial concentration of substance \(A\), and \(k\) is a positive constant. Determine the value of \(x\) at which the rate \(d x / d t\) of the reaction is maximum.

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{2}-6 x ; a=0, b=4 $$

Find all antiderivatives of the given function. 0

If \(C(x)\) is the cost of manufacturing an amount \(x\) of a given product and \(p\) is the price per unit amount, then the profit \(P(x)\) obtained by selling an amount \(x\) is $$P(x)=p x-C(x)$$ (Notice that there is a loss if \(P(x)\) is negative.) a. If \(C(x)=c x\) and \(c

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x-3 x^{2} ; a=-1, b=3 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} 4 /(2-x) $$