 # The temperature T(x) at each point x on the surface of Mars (a sphere) is a continuous function. Show that there is a point x on the surface such that T(x)=T(−x) Baladdaa9 2022-07-23 Answered
The temperature $T\left(x\right)$ at each point $x$ on the surface of Mars (a sphere) is a continuous function. Show that there is a point $x$ on the surface such that $T\left(x\right)=T\left(-x\right)$
(Hint: Represent the surface of Mars as $\left\{x\in {\mathbb{R}}^{3}:||x||=1\right\}$.)
Consider the function $f\left(x\right)=T\left(x\right)-T\left(-x\right)$
So.....
I consider an unit sphere is locating at the origin of a $xyz$-plane.
As $||x||=1$, I can say with $radius=1=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$
To find there is a point $T\left(x\right)=T\left(-x\right)$, we use the formula $f\left(x\right)=T\left(x\right)-T\left(-x\right)$ and show somehow $f\left(x\right)$ will equal to 0 ???
It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using $ijk$ plane) on the lower hemisphere
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Hint: Pick some point $x$. If you are lucky, $f\left(x\right)=0$, but probably this does not occur. In the other case, what can you say about the relation between $f\left(x\right)$ and $f\left(-x\right)$? Now connect $x$ and $-x$ using a path along the sphere...

We have step-by-step solutions for your answer! Bernard Boyer
If $f\left(x\right)=T\left(x\right)-T\left(-x\right)$ is zero, we are done. Assume there does not exist an $x$ that makes $f\left(x\right)$ zero. Then $f\left(x\right)>0$ or $<0$ for all $x$. If say $f\left(x\right)>0$ is never satisfied, then replacing $x$ by $-x$ we get a contradiction. Similarly if $f\left(x\right)<0$ never holds. So there must be points for which $f\left(x\right)>0$ and points for which $f\left(x\right)<0$. The intermediate value theorem then implies there must exist a point $y$ at which $f\left(y\right)=0$ (contradiction).

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