Question

The function $f\left(x\right)=\pi x^3-\frac{3\pi }{2}\left(a+b\right)x^2+3\pi abx$ has a local minimum at $x=a,$ then the values $a$ and $b$ can take are

• A. $a=\pi ,b=e$
• B. $a=e,b=\pi$
• C. $a=b=\pi$
• D. $a=b=e$

Answer 1

Answer: (B)

$f^{{}^{\prime }}\left(x\right)=3\pi \left(x^2-\left(a+b\right)x+ab\right)$
$=3\pi \left(x-a\right)\left(x-b\right)$

i.e. $f\left(x\right)$ has a local maximum at $x=a,$ if $a<b$