Question:

<b>Zeros of Derivative on Positive Reals</b><br> $y = x(x+2)\cdot x^3(3x+4) = x^4(x+2)(3x+4)$.<br> $y = x^4(3x^2+10x+8)$.<br> Roots of $y$: $x=0$ (multiplicity 4), $x=-2$, $x=-4/3$. On $(0,\infty)$, $y>0$ always (no real roots in $(0,\infty)$).<br> Since $y>0$ on $(0,\infty)$ and $y\to+\infty$ as $x\to\infty$, $y$ is increasing on $(0,\infty)$ with no local max/min.<br> $y'=0$ has no solutions in $(0,\infty)$. <b>Answer: 0</b><br> <b>Key concept:</b> Analyze sign of $y$ and behaviour; if $y>0$ and increasing everywhere on $(0,\infty)$, derivative has no zero there.<br> <b>Trap:</b> Computing $y'$ directly and trying to solve a degree-7 polynomial equation.