Nonlinear ProgrammingThe largest interval (a, b) of k ∈ R,for which the point (0, 0) is the critical point of the function f(x, y) = x2 + kxy + yans.

The sum of squares of all possible values of k, for which area of the region bounded by the parabolas 2y2=kx and ky2=2(y−x) is maximum, is equal to :

Let  f(x,y)=(cosx+ysinx)2. Let the maximum value of f(x,y) over all values of x for a given fixed value of y is called g(y). Let the smallest positive value x which achieves this maximum value of f(x,y) for a given y be h(y). If ∫2+√31h(y)g(y)dy=π2K, then value of K is equal to

The least value of k for which f(x) = x2 + kx +1 is increasing on (1, 2), is

[5 marks] Locate all relative maxima, relative minima and saddle points of thefunction f (x, y) = xy − x3 − y2.

Use the graph to state the absolute and local maximum and minimum values of the function. (Assume each point lies on the gridlines. Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)A function with two curves, three closed points, and three open points is graphed on the x y coordinate plane. The first curve begins at the closed point (0, 2) increases to the open point (1, 4) decreases to (2, 2) increases to (4, 5) decreases to (5, 3) then turns sharply and increases to the closed point (6, 4). The second curve begins at the open point (6, 3) and decreases to the open point (7, 1). The closed point (1, 3) is graphed.absolute maximum value 5 absolute minimum value 2 local maximum value(s) 5 local minimum value(s) 2,3