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Q.
If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to:
$ f^{\prime}(x)=x^2+2 b+a x $
$ g^{\prime}(x)=x^2+a+2 b x $
$ (2 b-a)-x(2 b-a)=0 $
$ \therefore x=1 $ is the common root
Put $x=1 \text { in } f^{\prime}(x)=0 \text { or } g^{\prime}(x)=0 $
$ 1+2 b+a=0 $
$ 7+2 b+a=6$