slimbluejays
New member
- Joined
- Jan 31, 2021
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For the function [MATH]F(\mathbf{x})=x^{a_1}_1x^{a_2}_2 \ldots x^{a_n}_n[/MATH] defined on the set [MATH]S=\{\mathbf{x}=(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1>0, \ldots ,x_n>0\}[/MATH] with [MATH]a_1,a_2,\ldots,a_n > 0[/MATH] and [MATH]a_1+a_2+\ldots+a_n=3[/MATH], I want to:
(i) Prove that [MATH]F(t\mathbf{x}) = t^3F(\mathbf{x})[/MATH] for every positive scalar [MATH]t[/MATH] and every [MATH]x \in S[/MATH].
I was able to work out this part by substituting every [MATH]x^{a_1}_1x^{a_2}_2 \ldots x^{a_n}_n[/MATH] with [MATH]tx^{a_1}_1tx^{a_2}_2 \ldots tx^{a_n}_n[/MATH].
(ii) Show that [MATH]\mathbf{x} \cdot \nabla F(\mathbf{x}) = 3F(\mathbf{x})[/MATH] at every [MATH]\mathbf{x}[/MATH], where [MATH]\nabla F(\mathbf{x}) = (\frac{\delta F(\mathbf{x})}{\delta x_1},\ldots,\frac{\delta F(\mathbf{x})}{\delta x_n})[/MATH].
I was able to work out that [MATH]\mathbf{x} \cdot \nabla F(\mathbf{x}) = a_1x^{a_1}_1 + \ldots + a_nx^{a_n}_n[/MATH] but got no further. Need help with this part!
(iii) Determine whether [MATH]F(\mathbf{x})[/MATH] is concave in [MATH]\mathbf{x}[/MATH] on the set [MATH]\mathbf{x}[/MATH].
My first thought was to use a Hessian matrix but that would be too tedious for this function. Is there any better method?
(i) Prove that [MATH]F(t\mathbf{x}) = t^3F(\mathbf{x})[/MATH] for every positive scalar [MATH]t[/MATH] and every [MATH]x \in S[/MATH].
I was able to work out this part by substituting every [MATH]x^{a_1}_1x^{a_2}_2 \ldots x^{a_n}_n[/MATH] with [MATH]tx^{a_1}_1tx^{a_2}_2 \ldots tx^{a_n}_n[/MATH].
(ii) Show that [MATH]\mathbf{x} \cdot \nabla F(\mathbf{x}) = 3F(\mathbf{x})[/MATH] at every [MATH]\mathbf{x}[/MATH], where [MATH]\nabla F(\mathbf{x}) = (\frac{\delta F(\mathbf{x})}{\delta x_1},\ldots,\frac{\delta F(\mathbf{x})}{\delta x_n})[/MATH].
I was able to work out that [MATH]\mathbf{x} \cdot \nabla F(\mathbf{x}) = a_1x^{a_1}_1 + \ldots + a_nx^{a_n}_n[/MATH] but got no further. Need help with this part!
(iii) Determine whether [MATH]F(\mathbf{x})[/MATH] is concave in [MATH]\mathbf{x}[/MATH] on the set [MATH]\mathbf{x}[/MATH].
My first thought was to use a Hessian matrix but that would be too tedious for this function. Is there any better method?