SATHEE: Indefinite Integration 1 Question 75

Indefinite Integration 1 Question 75

76. A cubic $f (x)$ vanishes at $x = - 2$ and has relative minimum / maximum at $x = - 1$ and $x = 1 / 3$ .

If $\int_{- 1}^{1} f (x) d x = 14 / 3$ , find the cubic $f (x)$ .

$(1992, 4$ M)

Show Answer

Answer:

Correct Answer: 76. $f (x) = x^{3} + x^{2} - x + 2$

Solution:

Since, $f (x)$ is a cubic polynomial. Therefore, $f^{'} (x)$ is a quadratic polynomial and $f (x)$ has relative maximum and minimum at $x = \frac{1}{3}$ and $x = - 1$ respectively, therefore, -1 and $1 / 3$ are the roots of $f^{'} (x) = 0$ .

$\begin{aligned} ∴ f^{'} (x) & = a (x + 1) x - \frac{1}{3} = a x^{2} - \frac{1}{3} x + x - \frac{1}{3} \\ = a x^{2} + \frac{2}{3} x - \frac{1}{3} \end{aligned}$

Now, integrating w.r. t. $x$ , we get

$f (x) = a \frac{x^{3}}{3} + \frac{x^{2}}{3} - \frac{x}{3} + c$

where, $c$ is constant of integration.

Again, $f (- 2) = 0$

$\begin{aligned} f (- 2) & = a - \frac{8}{3} + \frac{4}{3} + \frac{2}{3} + c \\ 0 & = a \frac{- 8 + 4 + 2}{3} + c \\ \Rightarrow & 0 & = \frac{- 2 a}{3} + c \Rightarrow c = \frac{2 a}{3} \\ ∴ & f (x) & = a \frac{x^{3}}{3} + \frac{x^{2}}{3} - \frac{x}{3} + \frac{2 a}{3} = \frac{a}{3} (x^{3} + x^{2} - x + 2) \end{aligned}$

Again, $\int_{- 1}^{1} f (x) d x = \frac{14}{3}$

[given]

$\begin{aligned} \Rightarrow \int_{- 1}^{1} \frac{a}{3} (x^{3} + x^{2} - x + 2) d x = \frac{14}{3} \\ \Rightarrow \int_{- 1}^{1} \frac{a}{3} (0 + x^{2} + 0 + 2) d x = \frac{14}{3} \end{aligned}$

$[∵ y = x^{3}$ and $y = - x$ are odd functions $]$

$\Rightarrow \frac{a}{3} 2 \int_{0}^{1} x^{2} d x + 4 \int_{0}^{1} 1 d x = \frac{14}{3}$

$\Rightarrow \frac{a}{3} \frac{2 x^{3}}{3} + 4 x = \frac{14}{3}$

$\begin{aligned} \Rightarrow & \frac{a}{3} \frac{2}{3} + 4 & = \frac{14}{3} \Rightarrow & \frac{a}{3} \frac{14}{3} = \frac{14}{3} \\ \Rightarrow & a & = 3 \end{aligned}$

Hence,

$f (x) = x^{3} + x^{2} - x + 2$

Indefinite Integration 1 Question 75

76. A cubic $f (x)$ vanishes at $x = - 2$ and has relative minimum / maximum at $x = - 1$ and $x = 1 / 3$ .

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Indefinite Integration 1 Question 75

76. A cubic f(x) vanishes at x=−2 and has relative minimum / maximum at x=−1 and x=1/3.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

76. A cubic $f (x)$ vanishes at $x = - 2$ and has relative minimum / maximum at $x = - 1$ and $x = 1 / 3$ .