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For no minima on the endpoint, I am thinking a using a polynomial with odd degree so that the both ends will go upward to positive and negative infinity respectively.

However, I have no idea how to have two maxima but no minima in between.

Please leave a hint, and much appreciated!

Hailey Chan
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3 Answers3

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If a function of one variable has two local maxima, it must have a local minimum in between.

A higher dimensional function could have two local maxima and no other critical points. Take for a example $$f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2.$$ This function has two local maxima at $(-1,0)$ and at $(1,2)$. You can write out the partial derivatives with respect to $x$ and $y$ and see that the only place they’re both zero is at the two local maxima.

Here’s a plot of the function: enter image description here

vitamin d
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    A function of one variable could have two maxima and no minima between if its domain need not be connected. [OP doesn't say domain must be all reals... or anything about the domain.] – coffeemath Mar 03 '21 at 17:25
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$f(x)=-x^{-2}$ has two maxima at $\pm \infty $ and no minimum.

Marius S.L.
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  • Can you make a different example not having its two maxes at $\pm \infty$ ? Because those are not real numbers. – coffeemath Mar 03 '21 at 17:39
  • The point is that you must have a point in between where the function is not defined. So you can simply put two "parabolas" next to each other without intersection. This condition rules out true parabolas, so their neighboring branches should be hyperbolas or similar. – Marius S.L. Mar 03 '21 at 18:04
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The function $f(x)=x^2-x^4$ has its maximum value of $f(c)=1/4$ attained at the two points $c=\pm \sqrt{1/2}.$ It has a local minimum at $x=0$ and no other local minima.

This function can be adjusted by restricting its domain to all reals except those in the closed interval $[-1/2,1/2].$ Calling the resulting restricted function $g(x)$ we now have an example of a twice differentiable function having two local maxes and no minimum. [Its graph does approach two non-attained low points at $(\pm 1/2, 3/8),$ but these are not technically local minima (not on the graph)]

coffeemath
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