Question

The real valued infinitely differentiable function g(x) is such that g(0) = 1, g'(0) = 2,and g"(0) = 3. Furthermore,g has the property that g(x) + $g^1$(x) + $g^2\left(x\right)$ + $g^3\left(x\right)$ = 0 and $g^n\left(x\right)$ denotes the nth derivative of g. Then,
(The notation[x] denotes the greatest integer that is less than or equal to x)

• A. $g^3\left(0\right)=-6$
• B. $g^3\left(0\right)=-8$
• C. [ $g^1(-1)+g^3(-1)$] = -12
• D. [$g^1(-1)+g^3(-1)$] = -11

Answer 1

The correct option is A $g^3\left(0\right)=-6$

It is given, that

$g\left(x\right)+g^1\left(x\right)+g^2\left(x\right)+g^3\left(x\right)=0$ where, $g^n\left(x\right)$ is the $nth$ derivative of $g\left(x\right)$

Put $x=0$ in the above given equation, we get

$g\left(0\right)+g^1\left(0\right)+g^2\left(0\right)+g^3\left(0\right)=0$

Given that, $g\left(0\right)=1,g^1\left(0\right)=2$ and $g^2\left(0\right)=3$

$\therefore 1+2+3+g^3\left(0\right)=0\Rightarrow g^3\left(0\right)=-6$