Derivatives - local extremes

Question

f(x) = x³ - 8x² + 5x

 

Compute algebraically the x-coordinates of the points on the graph of f where the tangent to this graph is horizontal. 

 

I know how to find the derivative:

 

f′(x) = 3x² - 16x + 5

 

And then to find the x - coordinates you solve f′(x) = 0

 

And this might be a really simple question, but I do not know how to solve the derivative for 0 and how to go about it.

 

Thank you so much.

Answer 1

0 = 3x² - 16x + 5

 

You find x by factoring or by using the quadratic formula.  This is basic algebra.  Calculus can be difficult if you can't do algebra.

 

When the leading coefficient (coefficient of the x² term) is not 1, factor using the "ac" method (http://virtualnerd.com/algebra-2/quadratics/solve-equations-by-factoring/factoring-strategies/factor-by-a-c-method) where a = 3, b = -16, c = 5.  You want two factors that multiply to give ac = (3)(5) = 15 and add to give b (=-16).  The factors are -15 and -1: (-15)(-1) = 15, -15 + (-1) = -16:

 

0 = 3x² - 16x + 5

0 = 3x² - 15x - x + 5

0 = 3x(x-5) - (x-5)

0 = (3x-1)(x-5)

 

x = 1/3 and 5