Question

What are the local extrema of `f(x)= x^3 - 9x^2 + 19x - 3`?

Answer 1

`f(x)_max =(1.37, 8.71)`
`f(x)_min =(4.63, -8.71)`

Explanation:

`f(x)= x^3-9x^2+19x-3`

`f'(x) = 3x^2-18x+19`

`f''(x) = 6x-18`

For local maxima or minima: `f'(x) =0`

Thus: `3x^2-18x+19 =0`

Applying the quadratic formula:

`x=(18+-sqrt(18^2-4xx3xx19))/6`

`x=(18+-sqrt96)/6`

`x=3+-2/3sqrt6`

`x~= 1.367 or 4.633`

To test for local maximum or minimum:

`f''(1.367) < 0 ->` Local Maximum

`f''(4.633) > 0 ->` Local Minimum

`f(1.367) ~= 8.71` Local Maximum
`f(4.633) ~= -8.71` Local Minimum

These local extrema can be seen on the graph of `f(x)` below.

graph{ x^3-9x^2+19x-3 [-22.99, 22.65, -10.94, 11.87]}