Answer:
Minimum at f(1) = -9½
Step-by-step explanation:
f(x) = ½ x² − x − 9
This is a parabola. Since the leading coefficient is positive, it faces upwards, meaning the vertex is a minimum. We can find the vertex using x = -b/(2a).
x = -(-1) / (2 × ½)
x = 1
f(1) = -9½
The minimum value is at (1, -9½).
We can also show this by completing the square to convert the equation to vertex form.
f(x) = ½ x² − x − 9
f(x) = ½ (x² − 2x) − 9
f(x) = ½ (x² − 2x + 1) − ½(1) − 9
f(x) = ½ (x − 1)² − 9½
Another way to find the minimum or maximum is with calculus.
f(x) = ½ x² − x − 9
f'(x) = x − 1
0 = x − 1
x = 1
When x < 1, f'(x) is negative. When x > 1, f'(x) is positive. Since f'(x) changes signs from negative to positive, f(1) is a minimum.
f(1) = -9½
