Questions: Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. f(x) = -(1/2) x^2 + 9x - 2 (a) The vertex is (Type an ordered pair, using integers or fractions.)

Solution Steps

To find the vertex of the quadratic function \( f(x) = -\frac{1}{2} x^{2} + 9 x - 2 \), we use the formula for the x-coordinate of the vertex:

\[ x = -\frac{b}{2a} \]

Substituting \( a = -\frac{1}{2} \) and \( b = 9 \):

\[ x = -\frac{9}{2 \cdot -\frac{1}{2}} = 9 \]

Next, we find the y-coordinate by substituting \( x = 9 \) back into the function:

\[ f(9) = -\frac{1}{2} (9)^{2} + 9(9) - 2 = -\frac{1}{2} \cdot 81 + 81 - 2 = -40.5 + 81 - 2 = 38.5 \]

Thus, the vertex is:

\[ \boxed{(9, 38.5)} \]

Since the coefficient \( a = -\frac{1}{2} \) is negative, the parabola opens downwards, indicating that the vertex represents a maximum value. The maximum value of the function is:

\[ \boxed{38.5} \]

Given that the parabola opens downwards and the maximum value is \( 38.5 \), the range of the function is:

\[ (-\infty, 38.5] \]

Final Answer