If a + b + c + d = 2, then the maximum value of (1 + a)( 1+ b)( 1 + c) (1 + d) is __________.

$\frac{81}{16}$

If a + b + c + d = 2,

then the maximum value of (1 + a)( 1+ b)( 1 + c) (1 + d)

Put the value of a , b, c, and d = equal to each other.

So,

4a = 2

a = \(\frac{1}{2}\)

Now put the value of a in the required equation,

(1 + a)( 1+ b)( 1 + c) (1 + d) = (1 + \(\frac{1}{2}\))( 1+ \(\frac{1}{2}\))( 1 + \(\frac{1}{2}\)) (1 + \(\frac{1}{2}\)) 

= \(\frac{3}{2}\) × \(\frac{3}{2}\) × \(\frac{3}{2}\) × \(\frac{3}{2}\) = \(\frac{81}{16}\)