Explanation:
To determine whether the quadratic function has a minimum or maximum, we need to look at the sign of the coefficient of the squared term. If the coefficient is positive, then the function has a minimum. If the coefficient is negative, then the function has a maximum. In this case, the coefficient of the squared term is positive (since it's a product of two positive factors), so the function has a minimum.
To find the minimum value of the function, we can use the formula:
x = -b / (2a)
where a is the coefficient of the squared term, b is the coefficient of the linear term, and x is the x-coordinate of the minimum (or maximum) point.
In this case, the quadratic function can be written in standard form as:
f(x) = 3x^2 + 42x + 72
where a = 3 and b = 42. Substituting these values into the formula, we get:
x = -42 / (2*3) = -7
So the minimum point occurs at x = -7. To find the corresponding minimum value of the function, we can substitute x = -7 into the original expression:
f(-7) = 3(-7+4)(-7+6) = 3*(-3)*(-1) = 9
Therefore, the quadratic function has a minimum value of 9, which occurs at x = -7.