extrema question?

Question

Find the extrema 9-x^1/3 (-1, 6)

Answer 1

1) find the derivative.
Are there any points on the interval where the value of the derivative is zero? If yes, this is a critical point.

2) take the second derivative (derivative of the derivative) and take its value at all critical points. If it is zero, then the critical point is an inflection (neither a maximum nor a minimum). If the second derivative is negative, then the critical point was a maximum.

3) If there are no maximum nor minimum on the interval, then they are at the border(s). Is the border part of the interval (from your description -- using round brackets -- maybe not).

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9-x^(1/3)
nine minus "cube root of x"
has an inflection point at 0.

derivative:
D(-x^(1/3) ) = -(1/3)x^(-2/3)
(the nine is a constant, so it does not contribute to the slope)

second derivative:
D(D(-x^(1/3) ) = (2/9)x^(-5/3)

One critical point is at x = 0 (D = 0)
However, it is an inflection point (DD = 0)

-x^(1/3) is positive when x is negative and
-x^(1/3) is negative when x is positive.

On either side of zero, the absolute value of x^(1/3) grows without bound, and without changing direction

There is no number where the cube root stops moving and reverses direction. This is confirmed by the fact that D(-x^1/3) never gets to zero (except at zero).

We only need to check the borders:
x = -1 then 9-x^(1/3)= +10
(x= 0 then 9-x^(1/3)= +9)
(x = 1 then 9-x^(1/3)= +8)
x= 6 then 9-x^(1/3)= +7.182879...


If -1 and 6 are part of the interval, then that is where the extrema are located.

If -1 and 6 are NOT part of the interval, then the function does not have an extremum.

It is bounded above by 10 (it cannot have a value greater than 10), but 10 itself is not part of the image (it cannot reach the value 10). However, there is no value below 10 that you can call the maximum.

For example, even if you claim that 9.99999 is a maximum, you can find a value of x (closer to -1) such that the function 9-x^(1/3) is worth 9.999999 (one more 9). You can keep on going like that forever. You can get as close as you want to 10.