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Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

The function ...

Question

The function $f (x) = 3 + 2 (a + 1) x + (a^{2} + 1) x^{2} - x^{3}$ has a local minimum at $x = x_{1}$ and local maximum at $x = x_{2}$ such that $x_{1} < 2 < x_{2}$ , then $a$ belongs to interval(s).

(- \infty, - \frac{3}{2})

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(- \frac{3}{2}, 1)

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(0, \infty)

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(1, \infty)

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Solution

The correct options are
A $(- \infty, - \frac{3}{2})$
D $(1, \infty)$
Since, $x_{1}$ & $x_{2}$ are local minima & local maxima of $f (x) = 3 + 2 (a + 1) x + (a^{2} + 1) x^{2} - x^{3}$
Therefore, $x_{1}$ & $x_{2}$ are the roots of the equation $f^{'} (x) = 0$
$\Rightarrow - 3 x^{2} + 2 (a^{2} + 1) x + 2 (a + 1) = 0$
Since, $2$ lies between the roots of the above quadratic equation
Therefore, $f^{'} (2) > 0$
$\Rightarrow - 12 + 4 (a^{2} + 1) + 2 (a + 1) > 0$
Therefore, $a$ belongs to $(- \infty, - \frac{3}{2})$ & $(1, \infty)$
Ans: A,D

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Similar questions

Q. If the function

f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1

has a local maximum at

x = x_{1}

and a local minimum at

x = x_{2}

such that

x_{2} = x_{1}^{2}

then the value of

^{'} a^{'}

equals.

Let $f (x) = ⎧ ⎨ ⎩ \begin{matrix} ∣ ∣ x^{2} - 3 x ∣ ∣ + a, 0 \leq x < \frac{3}{2} - 2 x + 3 x \geq \frac{3}{2} \end{matrix}$ If f(x) has a local maximum at x =.

Q. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

A function f(x) is increasing where f ′(x) > 0 and decreasing where f ′(x) < 0, then at x = x₁ function has its local minima. If f ′(x₁) = 0 and f ′′(x₁) > 0, similarly if at x = x₂, f′(x₂) = 0 and f ′′(x₂) < 0, then at x = x₂ function has its local maxima.

Given f(x) = x³ – 3x² – 9x + 3; x ∈ R

Choose the correct answer

एक फलन f(x) वर्धमान है जहाँ f ′(x) > 0 एवं ह्रासमान है जहाँ f ′(x) < 0, तब x = x₁ पर फलन का स्थानीय निम्निष्ठ है। यदि f ′(x₁) = 0 तथा f ′′(x₁) > 0, इसी प्रकार यदि x = x₂ पर, f′(x₂) = 0 तथा f ′′(x₂) < 0 है, तब x = x₂ पर फलन का स्थानीय उच्चिष्ठ है।

दिया है f(x) = x³ – 3x² – 9x + 3; x ∈ R

सही उत्तर का चयन कीजिए।

Q. f(x) has its local maxima at

प्रश्न - f(x) का स्थानीय उच्चिष्ठ है

Q. The function

f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1 = 0

has a local maximum at

x = α

, and a local minimum at

x = β

such that

β = α^{2}

then

a

is equal to:

Q. The total number of local maxima and local minima of the function

f (x) = {\begin{matrix} {(2 + x)}^{3}, & - 3 < x \leq - 1 x^{2 / 3}, & - 1 < x < 2 \end{matrix}

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The function f(x)=3+2(a+1)x+(a2+1)x2−x3 has a local minimum at x=x1 and local maximum at x=x2 such that x1<2<x2, then a belongs to interval(s).

The function $f (x) = 3 + 2 (a + 1) x + (a^{2} + 1) x^{2} - x^{3}$ has a local minimum at $x = x_{1}$ and local maximum at $x = x_{2}$ such that $x_{1} < 2 < x_{2}$ , then $a$ belongs to interval(s).