### Application Of Derivatives Question 7

#### Question 7 - 30 January - Shift 2

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{ax^2}{2}$ 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

(1) 4

(2) $\frac{3}{2}$

(3) 3

(4) 6

## Show Answer

#### Answer: (4)

#### Solution:

#### Formula: Maximum of function, Quadratic equations common roots

$f^{\prime}(x)=x^{2}+2 b+a x$

$g^{\prime}(x)=x^{2}+a+2 b x$

$(2 b-a)-x(2 b-a)=0$

$\therefore x=1$ is the common root

Put $x=1$ in $f^{\prime}(x)=0$ or $g^{\prime}(x)=0$

$1+2 b+a=0$

$7+2 b+a=6$