One way of proving that f x ≤ g x for all x in a given interval is to show that 0 ≤ g x − f x for all x in the interval; and one way of proving the latter inequality is to show that the absolute minimum value of g x − f x on the interval is nonnegative. Use this idea to prove the inequalities in these exercises. Prove that cos x ≥ 1 − x 2 / 2 for all x in the interval 0 , 2 π .
One way of proving that f x ≤ g x for all x in a given interval is to show that 0 ≤ g x − f x for all x in the interval; and one way of proving the latter inequality is to show that the absolute minimum value of g x − f x on the interval is nonnegative. Use this idea to prove the inequalities in these exercises. Prove that cos x ≥ 1 − x 2 / 2 for all x in the interval 0 , 2 π .
One way of proving that
f
x
≤
g
x
for all
x
in a given interval is to show that
0
≤
g
x
−
f
x
for all
x
in the interval; and one way of proving the latter inequality is to show that the absolute minimum value of
g
x
−
f
x
on the interval is nonnegative. Use this idea to prove the inequalities in these exercises.
Prove that
cos
x
≥
1
−
x
2
/
2
for all
x
in the interval
0
,
2
π
.
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