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

If y=f(x) and the function is known to attain a maximum point at x0, then at the point x0a.f’(x0) <0b.f’(x0) >0c.f’’(x0) >0d.f’’(x0) <0

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?

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.

For what interval of x-values is the curve y = f (x) concave up?

Suppose f(x) is a quadratic function. The slope of f(x) at the extremum point is a.0b.infinitec.negatived.positive