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STEP II 2007 question 7 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP II 2007 question 7 solution Since
, and Similarly,
which is negative for (i) First we note that Applying Jensen's inequality with
But since
from which the desired result immediately follows. (Note that since (ii) Setting, in Jensen's inequality,
Exponentiating both sides (note that since
as required. a) Setting, in
from which follows (note that we in this step use
as required. (Again, the usage of Jensen's inequality was valid, since b) Let Set, in [b](*)[/b],
(This inequality is valid whenever Now, this means
But and so the desired minimum value for the expression is -2. Solution by ukgea. |











in
,
is concave in this interval.
,
, (and indeed negative for
as well, the only problem being
is not defined there), and thus
since they are angles in a triangle.
,
and
yields
.
, the RHS equals
, and thus we have
are angles in a triangle,
and so our usage of Jensen's inequality was valid.)
and
, we get
is an increasing function, this preserves the inequality)
, we get
which was a requirement for the function to be concave)
.
,
, and we get
are positive, because then
are positive as well, which was a requirement for [b](*)[/b] to be valid)
.
attains this value at
, which means that this is the function's minimum. (By definition, since we have just proven that for all
, where
)





