If x1 and x2 are absicca of two points on the curve f(x)=x-x2 in the interval (0,1), then maximum value of expression (x1+x2)-(x12+x22) is
24done by bipin bhaiya in chatbox
its just an objective approach
15:46
b_k_dube.._45: find max value of x - x^2
b_k_dube.._45: it comes at x = 1/2
b_k_dube.._45: which is between
b_k_dube.._45: 0 and 1
b_k_dube.._45: so z1 and x2 are approximately 1/2 but not exactly equal to each other
[1][1]
66Note that
(x1+x2)-(x12+x22) = (x1-x12)+(x2-x22) = f(x1)+f(x2)
So we are looking for the maximum value of the sum of the ordinates.
It is easy to see that y=f(x) is downward opening parabola having roots 0 and 1 and maximum at x=1/2. The maximum value itself is 1/4. As such the given expression has a maximum value of 1/2 attained when both x1 and x2 are equal to 1/2.