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Calculate the greatest and least values of the functionf(x)=(x^4)/(x^8+2x^6-4x^4+8x^2+16) |
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Answer» Solution :`(1)/(f(x)) = (x^(4) + (16)/(x^(4))) + 2 (x^(2) + (4)/(x^(2))) - 4` Now, `A.M ge G.M` `IMPLIES x^(4) + (16)/(x^(4)) ge 8` and `x^(2) + (4)/(x^(2)) ge 4` `implies (1)/(f(x)) ge 12 implies f(x) le (1)/(12)` Again using `A.M ge G.M` we have `(2x^(6) + 8 x^(2))/(2) ge 4x^(4)` or `2 x^(6) + 8 x^(2) - 4 x^(4) ge 4 x^(4) ge 0` or `x^(8) + 2x^(6) - 4x^(4) + 8x^(2) + 16 lt 0` ALSO, `x^(4) ge 0` `implies (x^(4))/(x^(8) + 2x^(6) - 4x^(8) + 8x^(2) + 16) ge 0` `implies f(x) ge 0` HENCE, the greatest value is 1/12 and the least value is 0 |
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