Question

show that 3x 4 + 1 is o(x 4/2) and x 4/2 is o(3x 4 + 1)​

Answer 1

SOLUTION

TO PROVE

$\displaystyle\sf{3 {x}^{4} + 1 = O \bigg( \frac{ {x}^{4} }{2} \bigg) \: \: \: and \: \: \: \frac{ {x}^{4} }{2}= O \bigg( 3 {x}^{4} + 1 \bigg)}$

CONCEPT TO BE IMPLEMENTED

If f(x) & g(x) are two real valued function then f(x) = O(g(x)) if there exists a non zero real number c such that f(x) ≤ c g(x)

PROOF

$\displaystyle\sf{3 {x}^{4} + 1 }$

$\displaystyle\sf{ \leqslant 3 {x}^{4} + {x}^{4} }$

$\displaystyle\sf{ = 4 {x}^{4} }$

$\displaystyle\sf{ = 8. \frac{ {x}^{4} }{2} }$

$\displaystyle\sf{ \therefore \: \: 3 {x}^{4} + 1 = O \bigg( \frac{ {x}^{4} }{2} \bigg) \:}$

Again

$\displaystyle\sf{\frac{ {x}^{4} }{2}}$

$\displaystyle\sf{ = \frac{1}{6}.3 {x}^{4} }$

$\displaystyle\sf{ \leqslant \frac{1}{6} \bigg( 3 {x}^{4} + 1 \bigg) }$

$\displaystyle\sf{ \therefore \: \: \frac{ {x}^{4} }{2}= O \bigg( 3 {x}^{4} + 1 \bigg)}$

Hence proved

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