91Ó°ÊÓ

Consider the functions \(f(x)=x^{3}-7 x^{2}-5 x+4\) and \(f^{\prime}(x)=3 x^{2}-14 x-5\). Given that \(f\) is increasing where \(f^{\prime}(x)>0\) and \(f\) is decreasing where \(f^{\prime}(x)<0,\) find where \(f\) is increasing and where \(f\) is decreasing.

Pharmacy A doctor's prescription calls for the creation of pills that contain 12 units of vitamin \(\mathrm{B}_{12}\) and 12 units of vitamin E. Your pharmacy stocks two powders that can be used to make these pills: One contains \(20 \%\) vitamin \(\mathrm{B}_{12}\) and \(30 \%\) vitamin \(\mathrm{E},\) the other \(40 \%\) vitamin \(\mathrm{B}_{12}\) and \(20 \%\) vitamin E. How many units of each powder should be mixed in each pill?

The sum of two numbers is 7 and the difference of their squares is \(21 .\) Find the numbers.

The product of two numbers is 4 and the sum of their squares is 8 . Find the numbers.

Curve Fitting Find real numbers \(a, b,\) and \(c\) so that the graph of the function \(y=a x^{2}+b x+c\) contains the points \((-1,4),(2,3),\) and (0,1)

Use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places. \(\left[\begin{array}{rrrr}44 & 21 & 18 & 6 \\ -2 & 10 & 15 & 5 \\ 21 & 12 & -12 & 4 \\ -8 & -16 & 4 & 9\end{array}\right]\)

Express \(-5+5 i\) in polar form and in exponential form.

Curve Fitting Find real numbers \(a, b,\) and \(c\) so that the graph of the function \(y=a x^{2}+b x+c\) contains the points \((-1,-2),(1,-4),\) and (2,4)

The function \(f(x)=3+\log _{5}(x-1)\) is one-to-one. Find \(f^{-1}\).

Find the function \(y=a x^{2}+b x+c\) whose graph contains the points \((1,2),(-2,-7),\) and (2,-3).