\( f(x)=2 x^{2}+16 x-10 \) (a) The function's vertex is at the point \( (-4,-42) \) (Type an ordered pair, using integers or fractions.) (b) Determine whether the parabola has a maximum value or a minimum value and find the value. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a fraction.) A. The parabola opens downward and has a maximum value of B. The parabola opens upward and has a minimum value of -42 (c) The range of \( f(x) \) is \( [-42, \infty) \) (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (d) The function is increasing on ( \( -4, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) The function is decreasing on (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Let's analyze the function \( f(x) = 2x^2 + 16x - 10 \). ### (a) Vertex The vertex of a parabola given by the standard form \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Here, \( a = 2 \) and \( b = 16 \): \[ x = -\frac{16}{2 \cdot 2} = -\frac{16}{4} = -4 \] To find the y-coordinate of the vertex, substitute \( x = -4 \) back into the function: \[ f(-4) = 2(-4)^2 + 16(-4) - 10 = 2(16) - 64 - 10 = 32 - 64 - 10 = -42 \] Thus, the vertex is at the point \( (-4, -42) \). ### (b) Maximum or Minimum Value Since the coefficient of \( x^2 \) (which is \( a = 2 \)) is positive, the parabola opens upward. Therefore, it has a minimum value at the vertex. The minimum value is: \[ -42 \] So the correct choice is: **B. The parabola opens upward and has a minimum value of -42.** ### (c) Range of \( f(x) \) Since the parabola opens upward and the minimum value is \( -42 \), the range of \( f(x) \) is: \[ [-42, \infty) \] ### (d) Intervals of Increase and Decrease The function is decreasing on the interval to the left of the vertex and increasing on the interval to the right of the vertex. - The function is decreasing on \( (-\infty, -4) \). - The function is increasing on \( (-4, \infty) \). ### Summary of Answers (a) Vertex: \( (-4, -42) \) (b) Minimum value: **B. The parabola opens upward and has a minimum value of -42.** (c) Range: \( [-42, \infty) \) (d) The function is increasing on \( (-4, \infty) \). The function is decreasing on \( (-\infty, -4) \).