Explain, please?

If a function is always increasing, its slope is positive everywhere.

Take the derivative of the function to get a function for the slope at every point.

f'(x) = 3ax² + 2bx + c

That's a parabola. For it to be positive, it has to be above the x-axis. That means it has to never touch the axis (has no real zeros) and it must be upward facing.

A quadratic has no zeros if the discriminant is negative.

d = B² - 4AC < 0

A = 3a

B = 2b

C = c

d = (2b)² - 4(3a)(c) < 0

4b² - 12ac < 0

Divide both sides by 4:

b² - 3ac < 0

That satisfies the first part making sure it doesn't cross the x-axis (no real zeros).

In order to be upward facing, the leading coefficient (3a) must be positive.

3a > 0

Divide both sides by 3:

a > 0

Answer:

D. b² - 3ac < 0, a > 0