Class 11

Math

Algebra

Sequences and Series

Calculate the greatest and least values of the function

$f(x)=x_{8}+2x_{6}−4x_{4}+8x_{2}+16x_{4} $

$f(x)1 =(x_{4}+x_{4}16 )+2(x_{2}+x_{2}4 )−4$

$A.M≥G.M⇒x_{4}+x_{4}16 ≥8;x_{2}+x_{2}4 ≥4⇒f(x)1 ≥12⇒f(x)≤121 $

Again using $A.M≥G.M$

$22x_{6}+8x_{2} ≥4x_{4}⇒2x_{6}+8x_{2}−4x_{4}≥4x_{4}≥0⇒x_{8}+2x_{6}−4x_{4}−8x_{2}+16$

Also

$x_{4}≥0⇒x_{8}+2x_{6}−4x_{4}−8x_{2}+16x_{4} ≥0⇒f(x)≥0$

Hence the greatest value is $1/12$ and the least value is $0$