Here's my question and suggestion to solve it. Please let me know if I'm wrong.
Let $f$ be a continuous function in $\Bbb R$, where $f(a), f(b)$ are two Maxima and Minima points. $f(a), f(b)$ are the only Maxima and Minima Points of $f$ in $\Bbb R$.
Assuming $$f(a)<f(b), a<b$$ Prove that $f(a)$ is a local minimum point.
Since $f(a),f(b)$ are Maxima and Minima points, there is some $\varepsilon$ such that in the neighborhood of $a,b$, such that $$a,b\in[a-\varepsilon,b+\varepsilon]$$.
Since the function is continuous, according the Extreme Value Theroem, the function receives minimum and maximum in $[a-\varepsilon,b+\varepsilon]$. Since $f(a)<f(b)$, we conclude that $f(a)$ is a local minimum.
Am I wrong?