Determine whether f(x)= 1/2x^2-x-9 has a maximum or minimum value and find the value

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

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step-by-step explanation:

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