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

1 Answer
Sep 30, 2017

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.