Question

The least value of $f\left(x\right)=\frac{x^3}{3}-abx$ occurs at $x=$

• A. G.M of a,b
• B. A.M of a,b
• C. H.M of a,b
• D. A.G.M of a,b

Answer 1

The correct option is A G.M of a,b

$f\left(x\right)=\frac{x^3}{3}-abx$
$f^{\prime }\left(x\right)=x^2-ab$
Now, for maximum or minimum,
$f^{\prime }\left(x\right)=0$
Or
$x^2-ab=0$
Or
$x^2=ab$
Or
$x=\pm \sqrt{ab}$
Now
$f^{\prime \prime }\left(x\right)=2x$
At
$x=-\sqrt{ab}$
$f^{\prime \prime }\left(x\right)<0$.
Hence $f\left(x\right)$ attains a maximum at $x=-\sqrt{ab}$.
Similarly
$f^{\prime \prime }\left(x\right)>0$ at $x=\sqrt{ab}$.
Hence $f\left(x\right)$ attains a minimum at $x=\sqrt{ab}$.
Therefore the function attains a minimum at
$x=\sqrt{ab}$ which is the GM of a and b.