Use the Second Derivative Test to classify the critical points of f(x, y) = x^2 + 2y^2 - 4x + 4y + 6.

Define univariate analysis?

Average and marginal cost Consider the following cost functions.
a. Find the average cost and marginal cost functions.
b. Determine the average cost and the marginal cost when ${\displaystyle x=a}$.
c. Interpret the values obtained in part (b). ${\displaystyle C\left(x\right)=-0.04\times 2+100x+800,0\le x\le 1000,a=500}$

Find the integer a such that
${\displaystyle a\equiv 24\left(\text{mod}31\right)\text{and}-15\le a\le 15}$

A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the speciems for analysis
1) What the pmf of the number of granite specimens selected for analysis?
2) What is the probability that the number of granite specimens selected for analysis is greater than 7?

Determine which functions are polynomial functions. For those that are, identify the degree.
${\displaystyle f\left(x\right)=\frac{x^2+7}{3}}$

Using a table to evaluate composite functions Use the function values in the table to evaluate the following composite functions.
a. ${\displaystyle (f\cdot g)\left(0\right)}$ b. ${\displaystyle g\left(f(-1)\right)}$
c.${\displaystyle f\left(g\left(g(-1)\right)\right)}$,
$$\overline{)\begin{array}{}x& -2& -1& 0& 1& 2\\ f\left(x\right)& 0& 1& 3& 4& 2\\ g\left(x\right)& -1& 0& -2& -3& -4\end{array}}$$

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider ${\displaystyle f\left(t\right)=g\left(t\right)-h\left(t\right)}$ , where and are the position functions of the two runners.