Question

can someone please tell me the correct answer?​

Answer 1

Answer:

3rd

one is the answer

Step-by-step explanation:

Answer 2

Given,

$\longrightarrow f(x)=3x^3+2x^2-5x+\lambda$

Put $x=0.$

$\longrightarrow f(0)=\lambda$

Put $x=1.$

$\longrightarrow f(1)=\lambda$

If $f(x)$ has two distinct and real roots in $[0,\ 1],$ there should exist a real number $c\in(0,\ 1)$ such that $f'(c)=0.$

Taking first derivative of $f(x)$ wrt $x,$

$\longrightarrow f'(x)=9x^2+4x-5$

Put $x=c.$

$\longrightarrow f'(c)=9c^2+4c-5$

$\longrightarrow 9c^2+4c-5=0$

$\longrightarrow c=\dfrac{-4+\sqrt{16+180}}{18}\quad\quad\big[c\in(0,\ 1)\big]$

$\longrightarrow c=\dfrac{5}{9}$

Finding $f(c),$

$\longrightarrow f\left(\dfrac{5}{9}\right)=3\left(\dfrac{5}{9}\right)^3+2\left(\dfrac{5}{9}\right)^2-5\left(\dfrac{5}{9}\right)+\lambda$

$\longrightarrow f\left(\dfrac{5}{9}\right)=\lambda-\dfrac{400}{243}$

Case 1:-

$\longrightarrow f(0)\in[0,\ \infty)$

$\longrightarrow \lambda\in[0,\ \infty)\quad\quad\dots(1.1)$

and,

$\longrightarrow f(1)\in[0,\ \infty)$

$\longrightarrow \lambda\in[0,\ \infty)\quad\quad\dots(1.2)$

and,

$\longrightarrow f\left(\dfrac{5}{9}\right)\in(-\infty,\ 0)$

$\longrightarrow\lambda-\dfrac{400}{243}\in(-\infty,\ 0)$

$\longrightarrow\lambda\in\left(-\infty,\ \dfrac{400}{243}\right)\quad\quad\dots(1.3)$

Taking $(1.1)\land(1.2)\land(1.3),$

$\longrightarrow\lambda\in\left[0,\ \dfrac{400}{243}\right)\quad\quad\dots(1)$

Case 2:-

$\longrightarrow f(0)\in(-\infty,\ 0]$

$\longrightarrow \lambda\in(-\infty,\ 0]\quad\quad\dots(2.1)$

and,

$\longrightarrow f(1)\in(-\infty,\ 0]$

$\longrightarrow \lambda\in(-\infty,\ 0]\quad\quad\dots(2.2)$

and,

$\longrightarrow f\left(\dfrac{5}{9}\right)\in(0,\ \infty)$

$\longrightarrow\lambda-\dfrac{400}{243}\in(0,\ \infty)$

$\longrightarrow\lambda\in\left(\dfrac{400}{243},\ \infty\right)\quad\quad\dots(2.3)$

Taking $(2.1)\land(2.2)\land(2.3),$

$\longrightarrow\lambda\in\phi\quad\quad\dots(2)$

Taking $(1)\lor(2)$ we get,

$\longrightarrow\underline{\underline{\lambda\in\left[0,\ \dfrac{400}{243}\right)}}$