If functions f ( x )=frac{ x ^3}{3}+2 bx +frac{a x^2}{2} and g(x)=frac{x^3}{3}+a x+b x^2, a ≠ 2 b

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5. Continuity and Differentiation

hard

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

A

$4$

B

$\frac{3}{2}$

C

$3$

D

$6$

(JEE MAIN-2023)

Solution

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

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

$(2 b – a )- x (2 b – a )=0$

$\therefore x =1 \text { is the common root }$

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

$1+2 b + a =0$

$7+2 b + a =6$

Standard 12

Mathematics

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