Question

Show that the minimum value for the function f(x)=x²

+6x−20 occurs at x=−3. Show all necessary steps, explain and justify your answer. [5T/C]

Answer 1

The minimum value for the function f(x) = x² + 6x - 20 occurs at x = -3, which can be shown by completing the square to write the function in vertex form (x + 3)² - 29, revealing the vertex at (-3, -29).

Step-by-step explanation:

To show that the minimum value for the function f(x) = x² + 6x - 20 occurs at x = -3, we can use calculus or complete the square method. Here, we will use the latter.

First, let's express the function in vertex form:

1. Complete the square for the quadratic term and the linear term in the function:
   f(x) = (x² + 6x + 9) - 9 - 20 = (x + 3)² - 29.
2. x + 3 represents a shift to the left of the standard x² parabola by 3 units.
3. The vertex form (x + 3)² - 29 shows that the vertex of the parabola, and hence the minimum value of the function, is at (-3, -29) because the coefficient of x² is positive, indicating a parabola that opens upwards.

Therefore, the minimum value of the function f(x) occurs at x = -3.