Expert: Scotto - 3/20/2010

Question
1.Find the maximum value of the function:   

y = -x^2 + 8x - 4

A)8  B)22
C)16  D)12


Determine whether the following has a maximum or minimum value and then compute that value:

y = 2x^2 + 10x - 4
A)minimum;  16.5  B)maximum;  16.5
C)minimum;  -16  D)minimum;  -16.5


3. Determine the vertex coordinates for:

f(x) = 9 - 8x - x^2

A)(-4,5)  B)(4,25)
C)(-4,25)  D)(5,-4)


4. Determine the range for:

g(x) = 1/2(6 + x)(4 + x)

a)R = {y ≥ -1/2}  b)R = {all reals}
c)R = {-6,-4}  d)R = {y ≤ -1/2}


5. Solve:

5|n|-7 = 3

a)5 or -5  b)2 or -2
c).5 or -.5  d)No solution

Answer
1.Find the maximum value of the function y = -x² + 8x - 4.
The maximum can be found by taking the derivative and setting it equal to 0.
Thus, y' = -2x + 8, and y' is 0 at x = 4.  Put that into y = (-x+8)x - 4
and get y = (-4+8)4 - 4 = 12.  

A)8  B)22  C)16  D)12

Determine whether the following has a maximum or minimum value and then compute that value:
y = 2x² + 10x - 4

It can be seen that y' = 4x + 10 and y" = 4.
The second derivative is positive, so it is a minimum.
The maximum is where y' = 0, and that is at x = -5/2.
Putting that in gives 25/2 - 50/2 - 8/2 = -33/2, and that can be found below.

A)minimum;  16.5  B)maximum;  16.5
C)minimum;  -16  D)minimum;  -16.5


3. Determine the vertex coordinates for f(x) = 9 - 8x - x^2.
The vertex is where f'(x) = 0.  Note that f'(x) = -8 - 2x.
This is 0 at x=-4.  Find f(-4) to get the y value.
That is, its 9 + 32 - 16 = ...

A)(-4,5)  B)(4,25)  C)(-4,25)  D)(5,-4)


4. Determine the range for g(x) = (6 + x)(4 + x)/2.
It can be seen that this is really g(x) = (24 + 10x + x²)/2
This seen, the derivative can be seen to be g'(x) = (10 + 2x)/2 = 5 + x.
The critical point is where g'(x) = 0.

It can also be seen that g"(x) is 1, which is positive,
which means the critical value is a minimum.

Putting x = -5 in the equation will give a value that all y or less than.

a)R = {y ≥ -1/2}  b)R = {all reals}  c)R = {-6,-4}  d)R = {y ≤ -1/2}


5. Solve 5|n|-7 = 3.
Note that this is the same when 7 is added to both sides.  This gives 5|n|= 10.
Now divide both sides by 5 and get |n| = 2, so n=2 or n =-2.

a)5 or -5  b)2 or -2 c).5 or -.5  d)No solution


I've found the solutions (or close enough to get there).
You can find the answers with what was given.