Find all values of x for which the function is differentiable.

Question

given f(x) = (x³-8)/x²-4x-5m find, all values of x for which the function is differentiable.

 

How do I solve the question above? My teacher explained it today, but I was utterly and completely lost. If you could provide a relatively understandable explanation and a step-by-step solution, it would be much appreciated.

Thank you in advance!!!

Answer 1

Set the derivative of f(x) equal to zero.

 

f'(x) = [3x²(x² - 4x - 5) - (x³ - 8)(2x - 4)] / (x² - 4x - 5)² = 0

 

         [3x⁴ - 12x³ - 15x² - (2x⁴ - 4x³ - 16x + 32)] /  (x² - 4x - 5)² = 0

 

         (x⁴ - 8x³ - 15x² + 16x - 32) /  (x² - 4x - 5)² = 0

 

 

Next, try to to factor out the numerator.  Since the factored form of denominator is (x - 5)² (x + 1)², use this as a guide in the synthetic division process.  If synthetic division does not work, then just set the numerator equal to zero and solve for x.

 

Note:    x=4     and       -1         are vertical asymptotes.

 

 

 

 

x⁴ - 8x³ - 15x² + 16x - 32 = 0

 

Solve for x using the rational root theorem.