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For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}.
For a polynomial f, let f ' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf' + mf"), where f ∈ S, is _______
For a polynomial f, let f ' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf' + mf"), where f ∈ S, is _______
Correct answer is '5.00'. Can you explain this answer?
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Verified Answer
For a polynomial g(x) with real coefficient, let mg denote the number ...
ƒ(x) = (x2 – 1)2 h(x); h(x) = a0 + a1x + a2x2 + a3x3
Now, ƒ(1) = ƒ(–1) = 0
⇒ ƒ'(α) = 0 , α ∈ (-1, 1) [Rolle's Theorem]
Also, f'(1) = f'(-1) = 0
⇒ f'(x) = 0 has atleast 3 root, -1,α,1 with -1 < α < 1
⇒ f"(x) = 0 will have at leeast 2 root, say β, γ such that -1 < β < α < γ < 1 [Rolle's Theorem] So, min(mf'') = 2
and we find (mf' + mf'') = 5 for f(x) = (x2 - 1)2.
Now, ƒ(1) = ƒ(–1) = 0
⇒ ƒ'(α) = 0 , α ∈ (-1, 1) [Rolle's Theorem]
Also, f'(1) = f'(-1) = 0
⇒ f'(x) = 0 has atleast 3 root, -1,α,1 with -1 < α < 1
⇒ f"(x) = 0 will have at leeast 2 root, say β, γ such that -1 < β < α < γ < 1 [Rolle's Theorem] So, min(mf'') = 2
and we find (mf' + mf'') = 5 for f(x) = (x2 - 1)2.
Most Upvoted Answer
For a polynomial g(x) with real coefficient, let mg denote the number ...
We claim that $S$ has $7$ distinct elements.
First, note that $x^2$ has $1$ real root and $x^2+1$ has no real roots. Therefore, $m(x^2)=1$ and $m(x^2+1)=0$.
Next, let $a,b,c$ be real numbers with $a\neq 0$. Then the discriminant of $ax^2+bx+c$ is $b^2-4ac$. If this is negative, then $ax^2+bx+c$ has no real roots. If it is $0$, then $ax^2+bx+c$ has exactly $1$ real root. If it is positive, then $ax^2+bx+c$ has exactly $2$ real roots. Therefore, $m(ax^2+bx+c)$ is determined solely by the sign of $b^2-4ac$.
Case 1: $a>0$
We have $m(ax^2)=1$, $m(ax^2+1)=0$, $m(ax^2+x)=1$, $m(ax^2+2x+1)=1$, $m(ax^2-x)=2$, $m(ax^2-2x+1)=2$, and $m(ax^2+x+1)=2$.
Case 2: $a< />
We have $m(ax^2)=1$, $m(ax^2+1)=1$, $m(ax^2+x)=2$, $m(ax^2+2x+1)=2$, $m(ax^2-x)=1$, $m(ax^2-2x+1)=1$, and $m(ax^2+x+1)=0$.
Therefore, $S$ has $7$ distinct elements.
First, note that $x^2$ has $1$ real root and $x^2+1$ has no real roots. Therefore, $m(x^2)=1$ and $m(x^2+1)=0$.
Next, let $a,b,c$ be real numbers with $a\neq 0$. Then the discriminant of $ax^2+bx+c$ is $b^2-4ac$. If this is negative, then $ax^2+bx+c$ has no real roots. If it is $0$, then $ax^2+bx+c$ has exactly $1$ real root. If it is positive, then $ax^2+bx+c$ has exactly $2$ real roots. Therefore, $m(ax^2+bx+c)$ is determined solely by the sign of $b^2-4ac$.
Case 1: $a>0$
We have $m(ax^2)=1$, $m(ax^2+1)=0$, $m(ax^2+x)=1$, $m(ax^2+2x+1)=1$, $m(ax^2-x)=2$, $m(ax^2-2x+1)=2$, and $m(ax^2+x+1)=2$.
Case 2: $a< />
We have $m(ax^2)=1$, $m(ax^2+1)=1$, $m(ax^2+x)=2$, $m(ax^2+2x+1)=2$, $m(ax^2-x)=1$, $m(ax^2-2x+1)=1$, and $m(ax^2+x+1)=0$.
Therefore, $S$ has $7$ distinct elements.
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For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer?
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For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer?.
For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by S= {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer?.
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