Final answer:
The vertex of the quadratic function f(x)=3x²-24x+45 can be found using the vertex formula. The vertex is at (4, 93), and since the coefficient of the x² term is positive, the parabola opens upwards. Thus, the function is increasing for all x > 4, so the interval of increase is (4, ∞).
Step-by-step explanation:
To find the vertex of the quadratic function f(x)=3x²-24x+45, we can use the vertex formula h = -b/(2a) for the x-coordinate of the vertex, and substitute h back into the function to find the y-coordinate.
The function is in the form ax² + bx + c, where:
To find the x-coordinate of the vertex (h):
h = -b/(2a) = -(-24)/(2*3) = 24/6 = 4
We substitute h into the function to find the y-coordinate (k):
k = f(4) = 3(4)² - 24(4) + 45 = 48 + 45 = 93
The vertex of the quadratic function is (4, 93).
To determine the intervals where f is increasing, we note that since a > 0, the parabola opens upwards, so f is increasing for all x > 4. Therefore, the interval where f is increasing is (4, ∞).