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 intervals on which the function is increasing and the intervals on which the function is decreasing. \[ f(x)=x^{2}+10 x+24 \] (Type an integer or a fraction.) A. The parabola opens downward and has a maximum value of \( \square \) . B. The parabola opens upward and has a minimum value of -1 (c) What is the range of \( f(x) \) ? The range of \( f(x) \) is \( [-1, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (d) On what interval is the function increasing? The function is increasing on \( (-5, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) On what interval is the function decreasing? The function is decreasing on \( \square \). \( \square \) (Type your answer in interval notation. Use integers or fractions for any numbers in tire expression.)

We begin with the given function:   f(x) = x² + 10x + 24. Step (a): Find the vertex.   Complete the square:     f(x) = x² + 10x + 24       = (x² + 10x + 25) - 25 + 24       = (x + 5)² - 1.   Thus, the vertex is at (–5, –1). Step (b): Determine maximum or minimum value.   Since the coefficient of x² is positive (1 > 0), the parabola opens upward. This means the vertex is the minimum point.   Minimum value = –1. Step (c): Find the range.   The minimum output is –1 and the graph extends upward indefinitely.   Range: [–1, ∞). Step (d): Determine intervals of increase and decrease.   The vertex at x = –5 divides the parabola into two intervals.   • For x < –5, the function decreases.   • For x > –5, the function increases.   Thus, the function is:    • Increasing on (–5, ∞)    • Decreasing on (–∞, –5). Now we match these answers with the blanks: A. The parabola opens downward and has a maximum value of ☐.   (Answer: Incorrect premise; the parabola opens upward.) B. The parabola opens upward and has a minimum value of –1. C. The range of f(x) is [–1, ∞). D. The function is increasing on (–5, ∞) and decreasing on (–∞, –5). Thus, the answer to the blank for "On what interval is the function decreasing?" is:   (–∞, –5).