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Is-it-possible-to-find-how-many-real-roots-exist-in-the-equation-x-4-x-3-without-find-all-the-value-of-x-

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Question Number 23208 by Joel577 last updated on 27/Oct/17
Is it possible to find how many real roots exist in the equation x^4 + ∣x∣ = 3 without find all the value of x?
Isitpossibletofindhowmanyrealrootsexistintheequationx4+∣x∣=3withoutfindallthevalueofx?
Answered by mrW1 last updated on 28/Oct/17
x^4 +∣x∣≥0 f(x)=x^4 +∣x∣ is symmetric, i.e. f(−x)=f(x) f(x) is strictly increasing for x>0 and strictly decreasing for x<0 therefore x^4 +∣x∣=a (a<0) has no real root x^4 +∣x∣=0 has one real root, x=0 x^4 +∣x∣=a (a>0)has two real roots
x4+∣x∣⩾0f(x)=x4+∣x∣issymmetric,i.e.f(−x)=f(x)f(x)isstrictlyincreasingforx>0andstrictlydecreasingforx<0thereforex4+∣x∣=a(a<0)hasnorealrootx4+∣x∣=0hasonerealroot,x=0x4+∣x∣=a(a>0)hastworealroots

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