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anemone
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Prove that there is no root exists in the interval $(0,2)$ for a quartic function $k^4-10+k-3k^2$.
anemone said:Prove that there is no root exists in the interval $(0,2)$ for a quartic function $k^4-10+k-3k^2$.
We have $k(k-2)<0$. | Besides, we also have $k^3(k-2)<0$ |
Adding 1 to the inequality $0<k<2$ we get $1<k+1<3$. Or simply $k+1>0$. Thus, $k(k-2)(k+1)<0$ $k^3-k^2-2k<0$ $k^3<k^2+2k$ $2k^3<2k^2+4k$ (*) | Expanding the inequality we get $k^4-2k^3<0$ $k^4<2k^3$(**) |
agentmulder said:I have a slightly different approach...
A root of a polynomial equation is a value that, when substituted into the equation, makes the equation equal to zero. In other words, it is a solution to the equation. In this case, the polynomial equation is k^4 - 10k + k - 3k^2.
To prove that there is no root in the interval (0,2), we can use the Intermediate Value Theorem. This theorem states that if a continuous function changes sign over an interval, then it must have a root within that interval. However, since the given polynomial equation does not change sign over the interval (0,2), it has no root in that interval.
Yes, a graphical representation (or graph) of the polynomial equation k^4 - 10k + k - 3k^2 would be a curve that does not intersect the x-axis within the interval (0,2). This indicates that there is no root in that interval.
Proving that there is no root in the interval (0,2) is important because it helps us understand the behavior of the polynomial equation. It tells us that the equation does not have any solutions within that interval, which can be useful in solving other problems or analyzing the equation further.
Yes, the concept of proving no root in a given interval can be applied to other polynomial equations as well. By understanding the behavior of the polynomial equation and using mathematical theorems, we can determine the existence or non-existence of roots in a given interval for various polynomial equations.