Question

1)f(x)=2x^2 2)f(x)=-3x^3+3 3)f(x)=-1/2x^2+2

Answer 1

F'(x)=4x f'(x)=-6x^2 f'(x)=-x

Answer 2

1. The function f(x) = 2x^2 is a quadratic function. It is in the form of a parabola that opens upward. The coefficient 2 indicates that the parabola is relatively narrow compared to the standard x^2 parabola, which has a coefficient of 1. The vertex of this parabola is at the origin (0,0), and the axis of symmetry is the y-axis. The function values increase as the x-values increase.

2. The function f(x) = -3x^3 + 3 is a cubic function. It is a polynomial function of degree 3. The coefficient -3 indicates that the graph will be reflected vertically. The constant term +3 indicates that the graph will shift upward by 3 units. The function values will decrease as the x-values increase, and the graph may exhibit both positive and negative slopes.

3. The function f(x) = -1/2x^2 + 2 is also a quadratic function. The coefficient -1/2 indicates that the parabola opens downward, similar to the standard x^2 parabola but reflected vertically. The constant term +2 indicates a vertical shift upward by 2 units. The vertex of the parabola is at (0, 2), and the axis of symmetry is the y-axis. The function values decrease as the x-values move away from the vertex in both directions.