Min =Min
Consider the quadratic function If the minimum value of is the same as the minimum value of , what is the maximum value of the real number ?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The function f ( x ) = x 2 + 8 x + a = ( x + 4 ) 2 + ( a − 1 6 ) attains its minimum value of a − 1 6 at x = − 4 . The composite function:
f ( f ( x ) ) = ( x 2 + 8 x + a ) 2 + 8 ( x 2 + 8 x + a ) + a = [ ( x 2 + 8 x + a ) + 4 ] 2 + ( a − 1 6 )
has the same minimum value as f ( x ) when the quantity x 2 + 8 x + ( a + 4 ) = 0 ⇒ a = − x 2 − 8 x − 4 = − ( x + 4 ) 2 + 1 2 ⇒ a M A X = 1 2 .