If a^2 + b^2 = c^2 + d^2 = 6, then which of the following is/are

This relation reminds me of pythagorean theorem. We have two right triangles, one will legs a and b and hypotenuse \(\sqrt{6}\)
and other with legs c and d and hypotenuse \(\sqrt{6}\).

Looking at the extreme lengths of the legs, for each triangle, the legs may be anywhere from a little less than \(\sqrt{6}\) and a little more than 0 to both being equal at \(\sqrt{3}\).


I. \(ac+bd \leq 6\)

The maximum value of the product of two sides of different triangles will be \(\sqrt{3}*\sqrt{3}=3\)
The minimum value could be very very small since sqrt(6)*a very small value will give very small value. So maximum value of ac+bd is 6.
True.

II. \(ab+cd \leq 6\)

Again, a, b, c and d all can be \(\sqrt{3}\), so maximum value will be 6. Minimum can be very very small. True.

III. \(ad+bc \leq 6\)

This is no different from statement I. The equations are symmetrical about a and b and c and d so ac + bd is the same as ad + bc. The maximum value will be 6 and minimum will be very very small.

As discussed in statement II, statement IV cannot be true.

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