first reflection

okay, so my first reflection is going to be on the first derivative test. the first derivative test is used to find absolute maximums and minimums. To do these, first you take the derivative of the given equation, then you set it equal to zero and solve for x, then set up intervals using the x values, then plug in numbers within the intervals into the derivative to find where your relative maximums and minimums are, then you plug the x values into the original equation to find the absolute max's and min's.

Find the absolute maximums and minimums of the function f(x)=x^2-8x+4

First, you take the derivative of it:

f'(x)= 2x-8

Then you set it equal to zero and solve for x.

2x-8=0 2x=8 x=4

Then you set up the intervals

(-infinity, 4) U (4, infinity)

then you plug in a number found within the interval into the first derivative to find relative max's and min's.

(-infinity, 4)= negative number
(4, infinity)= positive number

so at x=4 there is a minimum

after that, you plug in that x value into the original equation to find the absolute max's and min's. (in this case it is only a min)

x=4 is the absolute minimum (4,-12)

There is a shortcut to finding absolute maximums and minimums, and that is using the second derivative test. all you do for that is take the first derivative, set it equal to zero and solve for x, then take the derivative of the first derivative (hence the name second derivative, then plug in the x value from the first derivative into the second derivative.

And that's all for my first reflection!!!