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Let f: $R\rightarrow R$ be a function such that $f(x)= x^3+x^2{f}'(1)+x{f}''(2)+{f}'''(3)$ , $x\epsilon R$ . Then f(2) equals :
Option 1)
-4
Option 2)
30
Option 3)
-2
Option 4)
8

Answers (1)

best_answer

Differential Equations -

An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable
$\left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )$

- wherein

eg:

$\frac{d^{2}y}{dx^{2}}- 3\frac{dy}{dx}+5x=0$

Order of a Differential Equation -

The order of a differential equation is order of highest order occuring in differential equation

- wherein

order of

$\frac{d^{2}y} {dx^{2}}+5=0$

is 2.

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

$=>f{}'(x)=3x^{2}+2xf{}'(1)+f{}''(2)$ ................................(1)

$=>f{}{}''(x)=6x+2f{}'(1)$ .....................................................(2)

$=>f{}{}''{}'(x)=6$ ..........................................................................(3)

put x=1 in equation (1)

$=>f{}'(1)=3+2f{}'(1)+f{}'{}'(2)$ .......................................(4)

put x=2 in equation (2)

$=>f{}''(2)=12+2f{}'(1)$ .....................................................(5)

from eqn (4) and (5)

$-3-f{}'(1)=12+2f{}'(1)$

$=>3f{}'(1)=-15$

$=>f{}'(1)=-5$ and $f{}'{}'(2)=2$

Now,

put x = 3 in eqn (3)

$f{}'{}'{}'(3)=6$

$\therefore f(x)=x^{3}-5x^{2}+2x+6$

$\therefore f(2)=8-20+4+6=-2$

Option 1)

-4

Option 2)

30

Option 3)
-2
Option 4)

8

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