Two real roots, I think because as x==> infinity , f(x) = x^4 + 12x -5 =0 ==> infinity , again for x==> (-inf.) ,
f(x) ==> infinity...because even degree ... and f'(x) possesses a single root (i.e., single minima for f(x) )....

First consider the behaviour at infinity on either side. The highest degree term is even,so this function has more of a valley behaviour. The derivative i.e. 4x^3+12=0 has only one real root. So f(x)=x^4+12*x-5 has one minima. If the value of f(x) at minima turns out to be positive. Then function has no real roots at all and if it is negative it has 2 real roots and if it is zero,then it has one root with multiplicity 2. In this case f(x) is negative at minima. So the function has 2 real roots.

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TopNewestTwo real roots, I think because as x==> infinity , f(x) = x^4 + 12x -5 =0 ==> infinity , again for x==> (-inf.) ,

f(x) ==> infinity...because even degree ... and f'(x) possesses a single root (i.e., single minima for f(x) )....

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That was a very good answer.

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First consider the behaviour at infinity on either side. The highest degree term is even,so this function has more of a valley behaviour. The derivative i.e. 4x^3+12=0 has only one real root. So f(x)=x^4+12*x-5 has one minima. If the value of f(x) at minima turns out to be positive. Then function has no real roots at all and if it is negative it has 2 real roots and if it is zero,then it has one root with multiplicity 2. In this case f(x) is negative at minima. So the function has 2 real roots.

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It has 2 real roots.

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