Inequalities

1) Show that for all positive reals a,b,c,d:-
1/a + 1/b + 4/c + 16/d>= 64/(a+b+c+d)

2) a,b,c are positive reals with abc=1. Prove that,
1/{a^3(b+c)} + 1/{b^3(c+a)} + 1/{c^3(a+b)}>= 3/2.

3) Solve the following inequality for real x:-
i) 813x-971<=163x^2
ii) 8x^2 + |-x| + 1 >0

2 Answers

1) from titu's lemma
1/a + 1/b + 4/c + 16/d >= (1+1+2+4)² / (a+b+c+d) = 64 / a+b+c+d

106

Asish Mahapatra ·2009-09-22 02:38:48

Q3. (ii) this is true for all x (think why?)

(i) rearranging 163x²-813x+971â‰¥0

This is a quadratic inequality. now can you solve

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