A function and its first and second derivatives are given. Use these to find eac

Question

A function and its first and second derivatives are given. Use these to find each of the following.

y = x1/3(x ? 8) y' = 4(x ? 2) 3x2/3 y'' = 4(x + 4) 9x5/3 A function and its first and second derivatives are given. Use these to find each of the following. Find the critical values. (Enter your answers as a comma-separated list.) x = Find the critical values. (Enter your answers as a comma-separated list.) Find the points of inflection. (Round the coordinates to one decimal place.) Find the points of inflection. (Round the coordinates to one decimal place.

Explanation / Answer

To obtain the relative maximum(s) and minimum(s), you have to use that derivative formula you have and set it equal to 0 or undefined. Since there is no denominator in the derivative, you can ignore the "set it equal to undefined" part.
So, for 3x^2 - 12x, you would first factor out the 3x, to get
3x(x-4). Next, set this equal to 0.
3x(x-4) = 0
You should get x=0, 4. These are called critical #'s.
Next, you make a number line of any length which include these 2 numbers as points on them:
<--O---O--->
Circle 1 is x=0, second circle is x=4.
Now test the intervals for the # line (test x<0, 0<x<4, and x>4).
To test, choose any value in the interval and plug it in to the derivative function. On the number line, mark whether the intervals are positive or negative.
+ - +
<--O--O-->
Circle 1 is x=0, second circle is x=4. Since f '(the derivative function) represents the slope of the tangent, you know there is a min when the slope changes from negative to positive, so 4 is your min. (Plug this into f(x) to get the y value of this point.) Also, when slope goes from + to -, you have a max, so 0 is the max. (Plug this in to f(x) to get the y-value of the point.)

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