a-maths

2x² + 6x + 8

= 2(x² + 3x + 4)

= 2[x² + 3x + (3/2)²] - 2(3/2)² + 8

= 2(x + 3/2)² + 7/2 > 0

So, f(x) = 1 / (2x² + 6x + 8) > 0 for all real values of x.


From the above, the minimum value of 2x² + 6x + 8 = 7/2 when x = -3/2

So, maximum value of f(x)

= 1 / (7/2)

= 2/7