Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f\left(g\left(x\right)\right)=8x^2-2x,$ and $g\left(f\left(x\right)\right)=4x^2+6x+1,$  then the value of f(2) + g(2) is……………..

 

Let f (x) = x⁶ + ax⁵ + bx⁴ + ax³ + dx² + ex + f

$\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^3}=1$    Non zero finite

So, d = e = f = 0

f (x)  = x⁶ + ax⁵ + bx⁴ + cx³

$\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^3}=1$ Non zero finite

f’ (x) = $6x^5+5a{x}^4+4b{x}^3+3x^2$

f’ (1) = 0

6 + 5a + 4b + 3 = 0

5a + 4b = - 9 . (i)

f’ (-1) = 0

-6 + 5a – 4b + 3 = 0 . (ii)

Solving (i) and (ii)

a  -3/5, b = -3/2

$\therefore f\left(x\right)=x^6+\left(\frac{-3}{5}\right)x^5+\left(\frac{-3}{2}\right)x^4+x^3$

$\therefore 5.f\left(2\right)=144$