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

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Answer 1 · 2017-09-30T08:31:07

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

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.