Let f : R [ ] R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x [ ] R.

Question

Let f : R $$ \to $$ R be a function such that  f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x $$ \in $$ R.  Then f(2) equals -

Accepted Answer

f(x) = x3 + x2f '(1) + xf ''(2) + f '''(3)

$$ \Rightarrow $$  f '(x) = 3x2 + 2xf '(1) + f ''(x)     . . . . . (1)

$$ \Rightarrow $$  f ''(x) = 6x + 2f '(1)     . . . . . . (2)

$$ \Rightarrow $$  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 equation (4) \& (5) :

-3 - f '(1) = 12 + 2f'(1)

$$ \Rightarrow $$  3f '(1) = - 15

$$ \Rightarrow $$  f '(1) = - 5 $$ \Rightarrow $$  f ''(2) = 2      . . . . .(2)

put x = 3 in equation (3) :

f ''' (3) = 6

$$ \therefore $$  f(x) = x3 - 5x2 + 2x + 6

f(2) = 8 - 20 + 4 + 6 = - 2

Solutions · 1