MathDB

2

Part of 1994 IMC

Problems(2)

IMC 1994 D1 P2

Source:

3/6/2017
Let f∈C1(a,b)f\in C^1(a,b), lim⁡x→a+f(x)=∞\lim_{x\to a^+}f(x)=\infty, lim⁡x→b−f(x)=−∞\lim_{x\to b^-}f(x)=-\infty and f′(x)+f2(x)≥−1f'(x)+f^2(x)\geq -1 for x∈(a,b)x\in (a,b). Prove that b−a≥πb-a\geq\pi and give an example where b−a=πb-a=\pi.
IMCreal analysis
IMC 1994 D2 P2

Source:

3/6/2017
Let f ⁣:R2→Rf\colon \mathbb R ^2 \rightarrow \mathbb R be given by f(x,y)=(x2−y2)e−x2−y2f(x,y)=(x^2-y^2)e^{-x^2-y^2}.
a) Prove that ff attains its minimum and its maximum.
b) Determine all points (x,y)(x,y) such that ∂f∂x(x,y)=∂f∂y(x,y)=0\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0 and determine for which of them ff has global or local minimum or maximum.
IMCreal analysis