If f ( x ) = x 3 - x 2 f ' ( 1 ) + x f ' ' ( 2 ) - f ' ' ' ( 3 ) , ? x R , then [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

Q.
If $f (x) = x^{3} - x^{2} f' (1) + x f'' (2) - f''' (3), x \in R,$ then [2023]

1 $2 f (0) - f (1) + f (3) = f (2)$

2 $f (3) - f (2) = f (1)$

3 $3 f (1) + f (2) = f (3)$

4 $f (1) + f (2) + f (3) = f (0)$

Ans.
(1)

Given, $f (x) = x^{3} - x^{2} f' (1) + x f'' (2) - f''' (3)$

$f' (x) = 3 x^{2} - 2 x f' (1) + f'' (2)$ ...(i)

$f'' (x) = 6 x - 2 f' (1)$ ...(ii)

$f''' (x) = 6 \Rightarrow f''' (3) = 6$ ...(iii)

From (iii),

$f'' (2) = 12 - 2 f' (1)$ ...(iv)

From (ii),

$f' (1) = 3 {(1)}^{2} - 2 f' (1) + f'' (2)$

$\Rightarrow f'' (2) = 3 f' (1) - 3$ ...(v)

$\Rightarrow$ $12 - 2 f' (1) = 3 f' (1) - 3$

$\Rightarrow f' (1) = 3;$ $f'' (2) = 12 - 6 = 6$ and $f' (1) = 3$

$f (x) = x^{3} - 3 x^{2} + 6 x - 6$ $\Rightarrow f (0) = - 6$

$f (1) = - 2, f (2) = 2, f (3) = 12$

$∴$ $2 f (0) - f (1) + f (3) = 2 = f (2)$

Solution

Q.
If $f (x) = x^{3} - x^{2} f' (1) + x f'' (2) - f''' (3), x \in R,$ then [2023]

1 $2 f (0) - f (1) + f (3) = f (2)$

2 $f (3) - f (2) = f (1)$

3 $3 f (1) + f (2) = f (3)$

4 $f (1) + f (2) + f (3) = f (0)$

Want Us to Email you About Special Offers & Updates?

Solution

Q. If f(x)=x3-x2f'(1)+xf''(2)-f'''(3), x∈R, then [2023]

1 2f(0)-f(1)+f(3)=f(2)

2 f(3)-f(2)=f(1)

3 3f(1)+f(2)=f(3)

4 f(1)+f(2)+f(3)=f(0)

Want Us to Email you About Special Offers & Updates?

Q.
If $f (x) = x^{3} - x^{2} f' (1) + x f'' (2) - f''' (3), x \in R,$ then [2023]

1 $2 f (0) - f (1) + f (3) = f (2)$

2 $f (3) - f (2) = f (1)$

3 $3 f (1) + f (2) = f (3)$

4 $f (1) + f (2) + f (3) = f (0)$