Question

The real valued infinitely differentiable function g(x) is such that $g\left(0\right)=1,g^{\prime }\left(0\right)=2$, and $g^{\prime \prime }\left(0\right)=3$. Furthermore, g has the property that $g\left(x\right)+g^1\left(x\right)+g^2\left(x\right)+g^3\left(x\right)=0$ and $g^n\left(x\right)$ denotes the nth derivative of g.
(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. $\left[g^1\right(-1)+g^3(-1\left)\right]=-12$
• D. $\left[g^1\right(-1)+g^3(-1\left)\right]=-11$

Answer 1

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

Given that

$g\left(0\right)=1g^{\prime }\left(0\right)=g^1\left(0\right)=2$

$g^{\prime \prime }\left(0\right)=g^2\left(0\right)=3$

and

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

put $x=0$

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

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

Option A is correct