Use the "mixed partials" check to see if the following differential equation is exact.
(3x^3 − 3y) dx + (−3x + 2y1) dy = 0
If it is exact find a function F(x, y) whose differential, dF(xy) is the left hand side of the differential equation. That is, level curves F(xy)=C are solutions to the differential equation.

The equation is exact

F(x,y) = 3x4/4  - 3xy -y2

Step-by-step explanation:

The step by step explanation and to ascertain the exactness of the differential equation is as shown in the attached file.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

4.5%

step-by-step explanation: