how to find the minimum and maximum value of 4(x-2)/(4x^2 +9)

Question

how to find the minimum and maximum value of y = 4(x-2)/(4x² +9)

 

Answer 1

Hi Julie,

 

So you want to find the minimum and maximum value of y = 4(x-2)/(4x^2 +9) = (4x-8) / (4x^2 +9)

 

Let's use the quotient rule:

 

y' = ((4x^2 +9)*(4)- ((4x-8)*(8x)) / ((4x^2 +9)^2

 

So y' = (16x^2 + 36 - 32x^2+64x) / (4x^2 +9)^2 = (-16x^2 +64x+36) / (4x^2 +9)^2 

 

so let's set the derivative equal to zero (we just need to set the numerator equal to zero):

 

-16x^2 +64x+36 = 0 

 

If you factor the quadratic equation, you have:

 

 

(4x-18)(4x+2) =0  from which you will get the roots x1=4.5   x2=-0.5

 

You could also use quadratic equation to find the roots.

 

Now we just need to do a sign test to see which one is a max and which one is a min:

 

 

 

 x                  -0.5                      4.5  

------------------------------------------------------------------

f'(x)    -----       0        ++++        0        -----

 

 

 

Since the derivative is negative when x< -0.5 and positive for x>-0.5. it implies that f(x) is decreasing until it reaches -0.5. and then it increases afterwards. So -0.5 is a min. And since the derivative is positive when x< 4.5 and negative when x>4.5, it implies that f(x) is first increasing, and then decreasing , and so 4.5 would be a max.

 

Now, if you want to know what the max and min values are, you just need to plug in these critical values, into the original function, and find them.

 

 

So f(-0.5) = -1         min

     f(4.5) = 0.1111   max

 

Note: It is a little hard to notice the function has a max at x=4.5 if you graph using a calculator. The function looks  like a horizontal line at one point (starting at x=3 and onward).  The min however, is obvious that occurs at -0.5.

 

 

 

 

 

 

Answer 2

So you need a min and max for 4(x-2)/(4x²+9)

 

Take the first derivative

 

4x² + 9 (4) - (4x-8) (8x)                         =    -16x² + 100

-----------------------------                              --------------------------

           (4x² + 9)                                            (4x² +9)

 

 

 

set this equal to and you will get + and -2.5

 

Here, you have found critical values. 

 

Take a value less than -2.5, a value between -2.5 and 2.5, and finally a value greater than 2.5.  The sign will determine whether the function is increasing or decreasing. 

 

I chose a value of -3.  this value says it is negative so the function is decreasing. 

 

Now I choose 0 which is between -2.5 and 2.5.  The value is positive. Therefore it is increasing from -2.5 to 2.5 and I choose a final value of 3 is negative. 

 

So the max is at x=2.5.