Hello! I can help you with that.
To find the extremum of the function k(x) = 5x² - 15x - 50 using the symmetry of its graph, we can follow these steps:
1. Determine the axis of symmetry:
The axis of symmetry for a quadratic function in the form of ax² + bx + c is given by x = -b / 2a. In this case, a = 5 and b = -15. So, the axis of symmetry is x = -(-15) / (2 * 5) = 15 / 10 = 1.5.
2. Find the vertex of the parabola:
The vertex of the parabola corresponds to the extremum point. Using the axis of symmetry (x = 1.5), substitute this value into the function to find the corresponding y-value:
k(1.5) = 5(1.5)² - 15(1.5) - 50
k(1.5) = 5(2.25) - 22.5 - 50
k(1.5) = 11.25 - 22.5 - 50
k(1.5) = -61.25
3. Classify the extremum:
Since the coefficient of x² is positive (5), the parabola opens upwards, which means the vertex at (1.5, -61.25) is the minimum point of the function.
Therefore, the extremum of the function k(x) = 5x² - 15x - 50 is a minimum at x = 1.5 with a value of -61.25.
