Question

$f\left(x\right)$ is a cubic polynomial with it's leading coefficient 'a'. $x=1$ is a point of extremum of $f\left(x\right)$ and $x=2$ is a point of extremum of $f\left(x\right)$. Then?

• A. The other point of extremum is at $x=3$
• B. The other point of extremum is at $x=0$
• C. If at $x=1$, $f\left(x\right)$ has a local maximum, $a>0$
• D. If at $x=1$, $f\left(x\right)$ has a local minimum, $a<0$

Answer 1

Answer: (C)

If at $x=1$, $f\left(x\right)$ has a local maximum, $a>0$

$\begin{array}{c}f\left(x\right)=9x^3+b{x}^2+cx+d\\ f^{\prime }\left(x\right)=3a{x}^2+2bx+c\\ x=1andx=2arerootsofearation\\ \Rightarrow 3a+2b+c=0\\ \Rightarrow 12a+4b+c=0\\ \Rightarrow b=-9a/2,c=69\\ f^{\prime }\left(x\right)=3x^2-9x+6\\ f^{\prime \prime }\left(x\right)=6x-9\\ forx=1pointofmaxima{f}^{\prime \prime }\left(x\right){}^{\prime }-v{e}^{\prime }\end{array}$

Option (c)