Final answer:
To find the minimum value of the quadratic function f(x) = 2x^2 + 16x + 30, use the vertex formula. The minimum occurs at the vertex x = -4, resulting in a minimum value of f(x) = -2.
Step-by-step explanation:
The student provided an equation f(x) = 2x2 + 16x + 30, which is a quadratic function. To find the minimum or maximum value, we can complete the square or use the vertex formula for a quadratic function. The quadratic function in the form f(x) = ax2 + bx + c has a vertex at (-b/(2a), f(-b/(2a))). Since 'a' is positive (2), the parabola opens upwards, and thus the vertex represents the minimum value of the function.
So, first calculate the x-coordinate of the vertex:
x = -b/(2a) = -16/(2*2) = -16/4 = -4
Now plug this back into the original equation to find the y-coordinate (which is the minimum value):
f(-4) = 2(-4)2 + 16(-4) + 30
f(-4) = 2(16) - 64 + 30
f(-4) = 32 - 64 + 30
f(-4) = -2
So, the minimum value of the function is when x = -4, and the minimum value is f(x) = -2.