If f(x) = \ array ll λ 2x + 3 , 0 ≤ x ≤ 3 + 12,3 < x ≤ 9 array is...

$∵ f ( x )$ is continuous at $x = 3$

$∴ LHL = RHL = f ( 3 )$

$\lambda\sqrt{2 ( 3 ) + 3} = \mu ( 3 ) + 12$ $⇒ \lambda = \mu +4$ ......(1)

Also $f ( x )$ is differentiable at $x = 3$

$∴$ at $x = 3$ $LHD = RHD$

$\dfrac{\lambda}{2\sqrt{2 ( 3 ) + 3}} \cdot 2 = \mu ⇒ \lambda = 3\mu$ ......(2)

By (1) and (2)

$\mu = 2, \lambda = 6$