Calculus Question

Question

Determine the extreme values of the function f(x) = x²-8x+9, where -1≤x≤5. What is the absolute minimum and the absolute maximum? 

Answer 1

You can also write the function in vertex form, since this is a parabola.

 

f(x) = (x² - 8x + 16) - 16 + 9

 

f(x) = (x - 4)² - 7

 

 

One minimum is f(4) = -7.    Compare this value to f(-1) and f(5).

Answer 2

Hi John.  

 

To find the extreme values, we take the derivative and set it equal to 0, and solve for the x's. These x's are called critical values, and we must plug them back into the original equation to find the y-coordinate of the mins and maxes.

 

When they ask for the absolute min and max, then you have to check the endpoints (x=-1 and x=5 in this case).  

 

Once you have found the y-values of all of the local mins and maxs (where the derivative =0) and know the y-values of the endpoints, then compare all of these y-values.   The largest y-value represents the ABS MAX.  The smallest y-value represents the ABS MIN.  A relative min can also me the absolute min and a relative max can also be the absolute max.  It just depends on which point (min, max, or endpoint) has the largest and smallest y-values.

 

Now, let's do it.

 

f'(x) = 2x - 8    Set this = 0 and get 

 

0 = 2x - 8, so 8 = 2x,  so x = 4 is the location of either a relative min or a relative max.  

We need it's y-coordinate, so put x=4 into the original funtion f(x) = x² - 8x + 9.

 

f(4) = -7    So there is a relative min at  (4, -7)

[How did I know it was a min?  The graph of x² + ...  is a parabola that points up like a "U", so the bottom of the parabola is a minimum.  Had it opened down (-x² looks like "∩") we would have just found the maximum (top of the frown) instead.]

Now check the endpoints:

f(-1) = 18

f(5) = -6

 

So there is a relative min at (4, -7)

There is an absolute min at (4, -7)

There is an absolute max at (-1,18)

 

When the question asks for the absolute minimum, they want the y-value only.  So the answer is "-7".

When the question asks for the absolute maximum, they want the y-value only.  So the answer is "18"

 

If they had asked "Where is the ...", then they would have wanted only the x-coordinate.