Final answer:
The gradient of the function f(x, y)=5/2 x³-6/5 y⁴ is obtained by computing its partial derivatives, resulting in the vector (15/2) * x² i - (24/5) * y³ j.
Step-by-step explanation:
To find the gradient of the function f(x, y)=5/2 x³-6/5 y⁴, we need to calculate its partial derivatives with respect to x and y. The gradient is a vector that points in the direction of the steepest ascent in the function and its magnitude represents the rate of increase in that direction.
The partial derivative of f with respect to x is Fx = (5/2) * 3 * x², which simplifies to Fx = (15/2) * x². Similarly, the partial derivative of f with respect to y is Fy = -6/5 * 4 * y³, which simplifies to Fy = -(24/5) * y³.
Therefore, the gradient of f(x, y) is the vector (15/2) * x² i + -(24/5) * y³ j, where i and j are the unit vectors in the direction of x and y respectively.