\( f(x)=2 x^{2}+16 x-10 \) (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 (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 (Type your answer in interval notation. Use integers or fractions for any numbers in the expression )

To analyze the function \( f(x) = 2x^2 + 16x - 10 \), we can follow the steps outlined in your questions. ### (a) Vertex The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = 2 \) and \( b = 16 \). Calculating the x-coordinate of the vertex: \[ x = -\frac{16}{2 \cdot 2} = -\frac{16}{4} = -4 \] Now, substituting \( x = -4 \) back into the function to find the y-coordinate: \[ f(-4) = 2(-4)^2 + 16(-4) - 10 = 2(16) - 64 - 10 = 32 - 64 - 10 = -42 \] Thus, the vertex is at the point: \[ \boxed{(-4, -42)} \] ### (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 of the function is: \[ \boxed{-42} \] ### (c) Range of \( f(x) \) Since the parabola opens upward and the minimum value is at \( -42 \), the range of \( f(x) \) is: \[ [-42, \infty) \] So, the answer in interval notation is: \[ \boxed{[-42, \infty)} \] ### (d) Increasing Interval The function \( f(x) \) is increasing when \( x \) is greater than the x-coordinate of the vertex. Since the vertex is at \( x = -4 \), the function is increasing for: \[ (-4, \infty) \] Thus, the answer in interval notation is: \[ \boxed{(-4, \infty)} \] In summary: - (a) Vertex: \((-4, -42)\) - (b) Minimum value: \(-42\) - (c) Range: \([-42, \infty)\) - (d) Increasing on: \((-4, \infty)\)