Question:
For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ in the set of polynomials with real coefficients defined by
$S =\left\{\left( x ^{2}-1\right)^{2}\left( a _{0}+ a _{1} x + a _{2} x ^{2}+ a _{3} x ^{3}\right): a _{0}, a _{1}, a _{2}, a _{3} \in R \right\} $
For a polynomial $f$, let $f'$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is
For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ in the set of polynomials with real coefficients defined by
$S =\left\{\left( x ^{2}-1\right)^{2}\left( a _{0}+ a _{1} x + a _{2} x ^{2}+ a _{3} x ^{3}\right): a _{0}, a _{1}, a _{2}, a _{3} \in R \right\} $
For a polynomial $f$, let $f'$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is
$S =\left\{\left( x ^{2}-1\right)^{2}\left( a _{0}+ a _{1} x + a _{2} x ^{2}+ a _{3} x ^{3}\right): a _{0}, a _{1}, a _{2}, a _{3} \in R \right\} $
For a polynomial $f$, let $f'$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is
Updated On: May 13, 2024
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Correct Answer: 5
Solution and Explanation
Then the minimum possible value of \(\left(m_{f'}+m_{f''}\right)\), where \(f \in S\), is \(\underline{5}.\)
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.