Question

What are the local extrema, if any, of `f (x) =(x^3−4 x^2-3)/(8x−4)`?

Answer 1

The given function has a point of minima, but surely doesnot have a point of maxima.

Explanation:

The given function is:

`f(x) = (x^3-4x^2-3)/(8x-4)`

Upon diffrentiation,

`f'(x) = (4x^3-3x^2+4x+6)/( 4*(2x-1)^2)`

For critical points, we have to set, f'(x) = 0.

`implies (4x^3-3x^2+4x+6)/( 4*(2x-1)^2) = 0`

`implies x ~~ -0.440489`

This is the point of extrema.

To check whether the function attains a maxima or minima at this particular value, we can do the second derivative test.

`f''(x) = (4x^3-6x^2+3x-16)/( 2*(2x-1)^3)`

`f''(-0.44) > 0`

Since the second derivative is positive at that point, this implies that the function attains a point of minima at that point.