In an A.P., the sixth term a _ 6 = 2 . If the a _ 1 a _ 4 a _ 5 is...

Given: ${a}_{6} = 2$. Also, ${a}_{1}{a}_{4}{a}_{5}$ is greatest.

$⇒ a + 5d = 2 . . . \left(i\right)$

Let, $y = {a}_{1}{a}_{4}{a}_{5}$

$⇒ y = a ( a + 3d ) ( a + 4d )$

$⇒ y = ( 2 - 5d ) ( 2 - 2d ) ( 2 - d )$

$⇒ y = \left(4 - 4d - 10d + 10{d}^{2}\right)\left(2 - d\right)$

$⇒ y = \left(4 - 14d + 10{d}^{2}\right)\left(2 - d\right)$

$⇒ y = 8 - 4d - 28d + 14{d}^{2} + 20{d}^{2} - 10{d}^{3}$

$⇒ y = 8 - 32d + 34{d}^{2} - 10{d}^{3}$

$⇒ {y}^{'} = - 32 + 68d - 30{d}^{2}$

$⇒ {y}^{'} = - 2\left(15{d}^{2} - 34d + 16\right)$

$⇒ {y}^{'} = - 2 ( 5d - 8 ) ( 3d - 2 )$

So,  the function attains maxima at $d = \dfrac{8}{5}$