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})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(0, \infty)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(1, \infty)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

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

Suggest Corrections

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}

Join BYJU'S Learning Program

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).