\( 1 \leftarrow \) 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. \( f(x)=-\frac{1}{2} x^{2}+5 x-1 \) (a) The vertex is (Type an ordered pair, using integers or fractions.)

To find the vertex of the quadratic function \( f(x) = -\frac{1}{2}x^2 + 5x - 1 \), we can use the vertex formula. The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -\frac{1}{2} \) and \( b = 5 \). Calculating the x-coordinate: \[ x = -\frac{5}{2 \times -\frac{1}{2}} = -\frac{5}{-1} = 5. \] Now substitute \( x = 5 \) back into the function to find the y-coordinate: \[ f(5) = -\frac{1}{2}(5^2) + 5(5) - 1 = -\frac{1}{2}(25) + 25 - 1 = -12.5 + 25 - 1 = 11.5. \] Thus, the vertex is \( (5, 11.5) \). Now, let’s dive into some interesting nuggets! Did you know that the vertex of a quadratic function is not just a point of interest in math? It represents the maximum or minimum value, depending on the direction of the parabola, and serves as a pivot for any graph sketching! Additionally, knowing how to find the vertex assists in problem-solving for optimization scenarios in fields like engineering and economics, where maximizing profits or minimizing costs are crucial tasks! The vertex provides quick insight into where these optimal values lie!