Convexitya Consider the function f(x) : R +R where pe R is a scalar:f(x) = |x|P.Show that, when p > 1, f(x) is convex. Show that, when 0 < p<1, f(x)is neither convex nor concave. Further, show that, when 0 0.

in the given question we have to check the concave and convex function but before check we know…

3 Convexity a Consider the function f(x) : R→ R where p e R is a scalar: S(x) = |x|". Show that, when p 2 1, f(x) is convex. Show that, when 0 < p < 1, f(x) is neither convex nor concave. Further, show that, when 0 < p< 1, f(x) is concave on the interval (-0,0] and is also concave on the interval [0, 0). b Let S be a nonempty, convex set in R". A function f(x) : S¬R is called quasi-convex if, for all 2 e [0, 1] and x1, x2 E S: f (Ax1 + (1– a )x2) s max [f(x1), f(x2)]. Show that a convex function is also quasi-convex. Using an example, show that a quasi-convex function is not necessarily convex. Hint Try the function in Part a.

3 Convexity a Consider the function f(x) : R→ R where p e R is a scalar: S(x) = |x|". Show that, when p 2 1, f(x) is convex. Show that, when 0 < p < 1, f(x) is neither convex nor concave. Further, show that, when 0 < p< 1, f(x) is concave on the interval (-0,0] and is also concave on the interval [0, 0). b Let S be a nonempty, convex set in R". A function f(x) : S¬R is called quasi-convex if, for all 2 e [0, 1] and x1, x2 E S: f (Ax1 + (1– a )x2) s max [f(x1), f(x2)]. Show that a convex function is also quasi-convex. Using an example, show that a quasi-convex function is not necessarily convex. Hint Try the function in Part a.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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