Dealing with a and b

Algebra Level 5

2 a 8 + 2 b 6 + a 4 − b 3 − 2 a 2 − 2 \large 2a^{8}+2b^{6}+a^{4}-b^{3}-2a^{2}-2

Find the minimum value of the expression for real numbers a , b a,b .


The answer is -2.75.

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Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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1 solution

Tom Engelsman
Aug 12, 2015

I've actually used partial derivatives taking:

d / d a = 16 a 7 + 4 a 3 − 4 a = 0 , d / d b = 12 b 5 − 3 b 2 = 0 d/da = 16a^7 + 4a^3 - 4a = 0, d/db = 12b^5 -3b^2 = 0

which yields the critical ordered-pairs:

( a , b ) = ( 0 , 0 ) ; ( 0 , ( 1 / 4 ) 1 / 3 ) ; ( 1 / 2 , 0 ) ; ( − 1 / 2 , 0 ) ; ( 1 / 2 , ( 1 / 4 ) 1 / 3 ) ; ( − 1 / 2 , ( 1 / 4 ) 1 / 3 ) . (a,b) = (0,0); (0, (1/4)^{1/3}); (1/\sqrt{2}, 0); (-1/\sqrt{2}, 0); (1/\sqrt{2}, (1/4)^{1/3}); (-1/\sqrt{2}, (1/4)^{1/3}).

Of these, the minimum value occurs at ( ± 1 / 2 , ( 1 / 4 ) 1 / 3 ) (\pm 1/\sqrt{2}, (1/4)^{1/3} ) , or − 2.75 \boxed{-2.75} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Even though you have obtained the right answer your solution is incomplete in a few ways. You have not included all the critical points of of both the equations.

Divyanshu Sharma - 5 years, 10 months ago

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