88th Problem 2016
Write the vertex of the following function as ( x , y ) :
f ( x ) = − 4 x 2 − 1 2 x + 9
Find x + y
Check out the set: 2016 Problems
The answer is 16.5.
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2 solutions
There are 3 methods of doing this: complete the square, the formula for turning points of quadratic functions (shown in the solution above) and this last method: differentiation.
f ( x ) = − 4 x 2 − 1 2 x + 9 f ′ ( x ) = − 8 x − 1 2 f ′ ′ ( x ) = − 8 < 0
Since f ′ ′ ( x ) < 0 , we know that the function's vertex is a maximum point (it doesn't matter in this question, but it might matter in other questions)
At turning points, f ′ ( x ) = 0
− 8 x − 1 2 = 0 8 x = − 1 2 x = − 8 1 2 = − 2 3 = − 1 . 5
f ( − 2 3 ) = − 4 ( − 2 3 ) 2 − 1 2 ( − 2 3 ) + 9 = − 9 + 1 8 + 9 = 1 8
The vertex is ( − 1 . 5 , 1 8 )
x = − 1 . 5 , y = 1 8 , x + y = − 1 . 5 + 1 8 = 1 6 . 5
The maximum value occurs when x = 2 × ( − 4 ) − ( − 1 2 ) = − 1 . 5
Substituting this value into f ( x ) we get the maximum value y = 1 8
Hence we get x + y = 1 6 . 5