- #1

chwala

Gold Member

- 2,675

- 352

- Homework Statement
- Let ##X## be a set of positive real numbers with infimum ## inf(x)=α>0##.

Let ##Y=\{\sqrt 2 -x^3| x\in X\} ##.

Find the supremum of ##Y## if it exists. Justify your answer.

- Relevant Equations
- Analysis

Refreshing on old university notes...phew, not sure on this...

Ok in my take, ##x>0##, and ##\dfrac{dy}{dx} = -3x^2=0, ⇒x=0## therefore, ##(x,y)=(0,\sqrt2)## is a critical point. Further, ##\dfrac{d^2y}{dx^2}(x=0)=-6x=-6⋅0=0, ⇒f(x)## has an inflection at ##(x,y)=(0,\sqrt2)##.

The supremum of ##Y## is ##\sqrt{2}.##

Ok in my take, ##x>0##, and ##\dfrac{dy}{dx} = -3x^2=0, ⇒x=0## therefore, ##(x,y)=(0,\sqrt2)## is a critical point. Further, ##\dfrac{d^2y}{dx^2}(x=0)=-6x=-6⋅0=0, ⇒f(x)## has an inflection at ##(x,y)=(0,\sqrt2)##.

The supremum of ##Y## is ##\sqrt{2}.##

Last edited: