Question

Chứng minh các bất đẳng thức sau :

a) \(5\left(x-1\right)< x^5-1< 5x^4\left(x-1\right)\), nếu \(x-1>0\)

b) \(x^5+y^5-x^4y-xy^4\ge0\), biết rằng \(x+y\ge0\)

c) \(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}< 5\), biết \(a,b,c\) cùng lớn hơn \(-\dfrac{1}{4}\) và \(a+b+c=1\)

Answer 1

a)\(x -1 >5 ⇔ x > 1 ⇒ x^4 > x^3 > x^2 > x > 1 \)

\(⇒ 5x^4 > x^4 + x^3 + x^2 + x + 1 > 5 \)

\(⇒ 5x^4 (x-1) > (x-1)( x^4 + x^3 + x^2 + x + 1) = x^5 -1 > 5 (x-1) \)

b)\(x^5 + y^5 – x^4y – xy^4 = (x + y)(x^4 – x^3y + x^2y^2 – xy^3 + y^4) – xy(x^3 + y^3) \)

\(= (x + y) [( x^4 – x^3y+ x^2y^2 – xy^3 + y^4) – xy(x^2 – xy + y^2)] \)

\(= (x + y) [(x^4+2x^2y^2+y^4) - 2xy(x^2+y^2)] \)

\(= (x + y) (x - y)^2(x^2 + y^2) ≥ 0 \)

c)\(\sqrt {4a + 1} + \sqrt {4b + 1} + \sqrt {4c + 1} )^2\)

\(= 4(a + b + c) + 3 + 2\sqrt {4a + 1} \sqrt {4b + 1} + 2\sqrt {4a + 1} \sqrt {4c + 1} + 2\sqrt {4b + 1} \sqrt {4c + 1} \)

\( \le 4(a + b + c) + 3 + (4a + 1) + (4b + 1) + (4a + 1) + (4c + 1) + (4b + 1) + (4c + 1) \)

\(\le 12(a + b + c) + 9 \le 21 \le 25\)