A tantalizing problem!!!
Calculus
Level
2
Let x be a number which exceeds its square by the greatest possible quantity.Then find the value of x.
The answer is 0.5.
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1 solution
This problem is asking you to find the minimum of x 2 − x . This can be done two ways.
The first method involves calculus. Set f ( x ) = x 2 − x . The extrema occur when the derivative is 0. Differentiating, f ′ ( x ) = 2 x − 1 . This function is 0 when x = 2 1 . Inspection reveals that this is indeed the minimum.
The other way is to notice that this is a simple quadratic function and as such, it's extrema will occur at it's vertex. The vertex of every parabola is found along it's axis of symmetry, 2 a − b . Plugging in b=-1 and a=1, we get the x-value of the vertex is 2 1 .