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I am working on a problem where I am at crossroads with ensuring whether or not the function $$f(x,y)=x\cdot \exp(-yx^{c})(1+2y) $$ has a unique maximum subject to $x\ge 1, 1\le y\le x$. (Here, $c$ is any positive constant.) Any help will be greatly appreciated. Thank you!

GA-Student
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  • What have you tried so far? – Dhanvi Sreenivasan May 23 '20 at 05:12
  • It is clear that the function is coercive. I tried to show that it is quasi-concave by definition, but this turns out to be tedious given the form of the function. If there is a specific method I should use here, would you kindly point that out. Thank you. – GA-Student May 23 '20 at 05:15
  • Have you tried the conventional optimization techniques for optimizing a multivariable function? Taking the gradient, finding critical points, etc. – paulinho May 23 '20 at 05:20

1 Answers1

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Let's look at one variable at a time. Consider $x$ as fixed, and lets see the effect of increasing y

$$f(y) = C(1+2y)\exp(-Ay)$$

Now, we can see that on increasing $y$, there will be a linear increase in one term and an exponential drop in the other term. Since exponential decay is much faster than linear rise, we would expect this term to decrease as $y$ increases, which can also be confirmed by taking derivatives or graphing

Hence, we should fix $y=1$

Now, for $x$, we have

$$f(x) = 3x\exp(-x^c)$$

If you were to check it's convexity, calculate $f"(x)$ for $x \geq 1$ and see what you get :)

Dhanvi Sreenivasan
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