<s>[INST] Minimum value of $p=3x+\frac{1}{15x}+5y+\frac{25}{y}+z+\frac{1}{36z},$ where $x,y,z\in \mathbb{R}^+$. 

Find the minimum value of $$p=3x+\frac{1}{15x}+5y+\frac{25}{y}+z+\frac{1}{36z},$$ where $x,y,z\in \mathbb{R}^+.$

Applying the AM-GM inequality,
$$
\begin{aligned}\frac{p}{6} & \geqslant\left(3x\cdot\frac{1}{15x}\cdot 5y\cdot\frac{25}{y}\cdot z\cdot \frac{1}{36z}\right)^{1/6} \\
\frac{p}{6} & \geqslant \left(\frac{5}{6}\right)^{1/3}\\
p & \geqslant 6\left(\frac{5}{6}\right)^{1/3}
\end{aligned}$$
$$\implies \text{The minimum value of the expression is } 6\left(\frac{5}{6}\right)^{1/3}
$$
Now, consider $f(x)=3x+\dfrac{1}{15x},\ g(y) = 5y+\dfrac{25}{y}$ and $h(z)=z+\dfrac{1}{36z}.$
$$\begin{aligned}f'(x) &= \frac{\mathrm d}{\mathrm dx}\left(3x+\frac{1}{15x}\right) = 3 - \frac{1}{15x^2}\\f''(x)&=\frac{2}{15x^3}\end{aligned}$$
At the critical points, $f'(x) = 0 \implies x = \dfrac{\pm1}{3\sqrt{5}}.$
$f''\left(\dfrac{1}{3\sqrt5}\right) > 0\implies f(x)$ has a local minima at $x = \dfrac{1}{3\sqrt5}.$
Similarly, the local minima of $g(y)$ is at $y = \sqrt5$ and the local minima of $h(z)$ is at $z=1/6.$
Substituting these values into the original expression, the minimum value of the expression comes out to be
$$
\begin{aligned}
p & = f\left(\dfrac{1}{3\sqrt5}\right)+g\left(\sqrt5\right)+h\left(\frac{1}{6}\right) \\
& = \frac{1}{3} + \frac{52}{\sqrt5}.
\end{aligned}
$$
The answer according to the AM-GM inequality is $\approx5.646$ and according to calculus is $\approx23.588$, which are way off. Also, the functions have only two points of inflection, one is the maxima (for values less than $0$) and the other is the minima (for values greater than $0$). As the question clearly states "for $x,y,z \in \mathbb{R}^+$", the "correct" answer should be $\approx23.588$, shouldn't it?
Why are the answers different? Also, is it possible to figure out the individual $x, y$ and $z$ values for which the expression has the minimum value (in case of AM-GM inequality)?

Note: This question is from a Test and the correct answer according to the "test creators" is $$6\left(\dfrac{5}{6}\right)^{1/3}.$$

$\text{Graph made using Desmos.}$
 [/INST] Hint: Rewrite the expression as 
\begin{eqnarray*}
\frac{1}{\sqrt{5} } \left( \sqrt{3} \sqrt[4]{5} x - \frac{1}{\sqrt{3} \sqrt[4]{5} x  } \right)^2 + \frac{2}{\sqrt{5}}
\\ +
5\sqrt{5}  \left( \frac{y}{ \sqrt[4]{5}} - \frac{ \sqrt[4]{5}}{y  } \right)^2 +10\sqrt{5} \\+ 
\frac{1}{6 } \left( \sqrt{6} z - \frac{1}{\sqrt{6} z   } \right)^2 + \frac{1}{3} .
\end{eqnarray*}
</s>