Question

Does the function f(x)=3x²−12x+1 have a minimum or a maximum? Identify its value.

a) Minimum at x=2
b) Maximum at x=2
c) Minimum at x=2 with a value of 1
d) Maximum at x=2 with a value of 1

Answer 1

The quadratic function f(x) = 3x² - 12x + 1 has a minimum, not a maximum, because the coefficient of x² is positive. Using the vertex formula, the minimum is at x=2. However, the minimum value is -11, which means the provided options are incorrect.

Step-by-step explanation:

The function f(x) = 3x² - 12x + 1 is a quadratic function, whose graph is a parabola. Since the coefficient of the x² term (3) is positive, the parabola opens upwards, which means that the function has a minimum, not a maximum. To find the value of this minimum, we can use the vertex formula, x = -b/2a, where a and b are coefficients from the function in the form ax² + bx + c.

Plugging the values from the function, we get x = 12/(2*3) = 2. Now, to find the minimum value of the function, we substitute x = 2 back into the equation: f(2) = 3(2)² - 12(2) + 1 = 12 - 24 + 1 = -11.

Therefore, the correct answer is a) Minimum at x=2 with the value of -11, which is not among the provided options.