Respuesta :

Answer:

f is increasing on (3, infinity)


Step-by-step explanation:

On which interval is the function f(x) = x^2 - 6x + 4 increasing?

1) Here's an approach that doesn't involve calculus:

Determine the vertex of this parabolic graph.  The x-coordinate of the vertex is x = -b / (2a), which here is x = -(-6) / (2*1), or x = 6/2, or x = 3.

The y-coord. of the vertex is f(3), which comes out to 3^2 - 6(3) + 4 = -5.  The vertex is then (3, -5).  Because the coeff. of the x^2 term is +, we know that the parabola opens up.  From x = -infinity to x = 3, the function is decreasing; from x = 3 to inf, the function is increasing.

2)  Here's the calculus approach:  The derivative of f(x) = x^2 - 6x + 4 is

f '(x) = 2x - 6.  We set this = to 0 and solve for x:  x = 3.

This tells us that the x-coordinate of the vertex is 3 (which we already knew).

Choose two test points:  x = 2 (on the left of x = 3) and x = 4 (on the right of x = 3).  Substitute 2 for x in f '(x) = 2x - 6; we get a negative result, which tells us that the function is decreasing on (-inf., 3).  Subst. 4 for x in f '(x) = 2x - 6; we get a positive result, which tells us that f is increasing on (3, infinity).