Consider the function f:R2→R𝑓:𝑅2→𝑅 defined byf(x,y)=x2+bxy+y2+alogcosh(x)𝑓(𝑥,𝑦)=𝑥2+𝑏𝑥𝑦+𝑦2+𝑎log⁡cosh⁡(𝑥)where a≥0𝑎≥0 and b𝑏 are real parameters. (a) If a=0𝑎=0, what is the smallest value of b𝑏 for which f𝑓 is convex?

1) Consider the following function, where A>0 and  0<α<1.Y=ALαThe function isa.strictly negative and concaveb.strictly negative and convexc.strictly positive and convexd.strictly positive and concave

Let f be a flow in a network, and let ˛ be a real number. The scalar flow product,denoted ˛f , is a function from V V to R defined by.˛f /.u; / D ˛  f .u; / :Prove that the flows in a network form a convex set. That is, show that if f1 and f2are flows, then so is ˛f1 C .1  ˛/f2 for all ˛ in the range 0  ˛  1.

A function f(x) is convex ifa.f’’ (x) >0b.f’’ (x) <0c.f’’ (x) = 0d.f’(x) >0

​The function 𝑓 is defined by 𝑓⁡(𝑥)=𝑎⁢sin⁡(𝑏⁢(𝑥+𝑐)+𝑑), for constants 𝑎, 𝑏, 𝑐, and 𝑑. In the 𝑥⁢𝑦-plane, the points (2,2) and (4,4) represent a minimum value and a maximum value, respectively, on the graph of 𝑓. What are the values of 𝑎 and 𝑑 ?

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.[0, 1](−∞, ∞)[-1,-1][1,0]This Question Is