Question

Which of the following could be not true if $f^{\prime \prime }\left(x\right)=x^{-1/3}$

• A. $f\left(x\right)=\frac{3}{2}{x}^{2/3}-3$
• B. $f\left(x\right)=\frac{9}{10}{x}^{5/3}-7$
• C. $f^{\prime \prime \prime }\left(x\right)=-\frac{1}{3}{x}^{-4/3}$
• D. $f^{\prime }\left(x\right)=\frac{3}{2}{x}^{2/3}+6$

Answer 1

The correct option is A $f\left(x\right)=\frac{3}{2}{x}^{2/3}-3$

$f^{\prime \prime }\left(x\right)=x^{-1/3}=\frac{d}{dx}(\frac{3}{2}{x}^{2/3}+C)$
So $f^{\prime }\left(x\right)=\frac{3}{2}{x}^{2/3}+c_1=\frac{d}{dx}(\frac{9}{10}{x}^{5/3}+c_{1}x+c_2)$
$\Rightarrow$ $f\left(x\right)=\frac{9}{10}{x}^{5/3}+c_{1}x+c_2$ ($c_1,c_2$ are constants.)
Thus option 'A' cannot be true.