Suppose the cube $x^3-p x+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the…

Suppose the cube $x^3-p x+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?
  1. The cubic has minima at $\sqrt{\frac{\mathrm{p}}{3}}$ and maxima at $-\sqrt{\frac{\mathrm{p}}{3}}$
  2. The cubic has minima at $-\sqrt{\frac{p}{3}}$ and maxima at $\sqrt{\frac{p}{3}}$
  3. The cubic has minima at both $\sqrt{\frac{p}{3}}$ and $-\sqrt{\frac{p}{3}}$
  4. The cubic has maxima at both $\sqrt{\frac{p}{3}}$ and $-\sqrt{\frac{p}{3}}$

Solution

Let $f(x)=x^3-p x+q$ Now for maxima/minima $ \begin{aligned} & f^{\prime}(x)=0 \\ & \Rightarrow 3 x^2-p=0 \\ & \Rightarrow x^2=\frac{p}{3} \\ & \therefore x=\pm \sqrt{\frac{p}{3}} \end{aligned} $

Asked in: JEE Main 2008

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