Let $a,b,c$ be distinct real numbers. What is the number of distinct real roots of the equation $(x-a)^3+(x-b)^3+(x-c)^3=0$?
- $1$,
- $2$,
- $3$,
- Depends on the value of $a,b,c$.
How can I solve this?
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2Did you try to think of Rolle's theorem? – Torsten Hĕrculĕ Cärlemän Jul 16 '13 at 08:36
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@Santosh are you trying to solve NET's question paper? – Mathronaut Jul 16 '13 at 15:33
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neeraj may be you thought – sam Jul 17 '13 at 04:04
3 Answers
17
Hint:
- Function $f(x) = x^3$ is strictly increasing in $x$.
- Let $f$ and $g$ be strictly increasing, then $x \mapsto f(x)+g(x)$ is strictly increasing as well.
I hope this helps ;-)
dtldarek
- 37,381
7
From the derivative, we see that there is a sum of three squares equalling zero, $$(x-a)^2+(x-b)^2+(x-c)^2 = 0$$
which implies that for a solution to exist $a=b=c$ Since, given that they are distint, the derivative has no roots. By Rolle's theorem, we may conclude that the original function may have atmost one root.
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3Continuing the answer: "...and since the function is a real polynomial one of odd degree then it has one solution" – DonAntonio Jul 16 '13 at 10:15
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@santoshkumar if you have a polynomial of degree n, then the derivative will be of degree n-1 and hence can admit at most n-1 roots, since the derivative has no roots then the polynomial should be a continuously increasing one(derivative is the slope, right?). Hence, only one root over reals – Torsten Hĕrculĕ Cärlemän Jul 16 '13 at 10:17
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i just dont understand by what he means by the existance of a root only if a=b=c. the correct answer is given by me please verify – Suraj M S Jul 16 '13 at 17:03
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@SurajM.S I have made a new explanation in my comment, please refer. I was talking about the roots of the derivative. – Torsten Hĕrculĕ Cärlemän Jul 16 '13 at 17:17
6
Let $y=(x-a)^3 +(x-b)^3 +(x-c)^3$
then $\dfrac{dy}{dx} =3( (x-a)^2 +(x-b)^2 +(x-c)^2 )$
therefore, $\dfrac{dy}{dx} > 0$
Also, $f(-\infty) = -\infty$ and $f( \infty ) = \infty$
i.e., $f(x)$ is strictly increasing from $-\infty$ to $\infty$
hence the function possesses $1$ real root for any values of $a, b ,c$
the number of answers is one.
