Functions in Algebra Examples

1. f(x) = 12 xe6x

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 3 − 2x + 4y − x2 − 4y2

Find the values of b such that the function has the given maximum value.

Maximum value: 62

f(x) = −x2 + bx − 19 

b =            (smaller value)

b =           (larger value)

What would be the graph of a density function with an area of z > -1.5. *

f(x)=X^2-2X-3

Consider the following.
$A=\left[\begin{array}{cc}-5& 9\\ -1& 1\end{array}\right]$
List the eigenvalues of $A$ and bases of the corresponding eigenspaces. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.)
smaller ${\displaystyle \lambda }$-value ${\displaystyle {\lambda }_1=}$ ___
smaller ${\displaystyle \lambda }$-value ${\displaystyle {\lambda }_2=}$ ___
Find an invertible matrix $P$ and a diagonal matrix $D$ such that ${\displaystyle p^{-1}AP=D}$. (Enter each matrix in the form $\left[\begin{array}{c}row 1\\ row 2\\ \dots \end{array}\right]$, where each row is a comma-separated list. If $A$ is not diagonalizable, enter NO SOLUTION.)
$(D,P)=\left(\_\_\_\right)$

Find a polynomial f(x) of degree 3 that has the following zeros. 8 ( multiplicity 2), -4.

a) Consider the following functional relation, f, defined as:
${\displaystyle f:R\to R,f\left(x\right)=x^2}$
Determine whether or not fis a bijection. If it is, prove it. If it is not, show why it is not 
b) Consider the set 
${\displaystyle F=\{y\mid y=a{x}^3+b\}}$
a, b being constants such that $a\ne 0$ and ${\displaystyle x\in R}$. 
Is ${\displaystyle F=R}$? If so, prove it. If not, explain in details why it is not the case.

Consider the sunction f(x)-x^1/2 and the function g, shown below