Question

Let $f, g$ and $h$ are differentiable function such that $g(x)=f(x)-x$ and $h(x)=f(x)-x^3$ are both strictly increasing functions, then the function $F(x)=f(x)-\frac{\sqrt{3} x^2}{2}$ is

• A. strictly increasing $\forall x \in R$
• B. strictly decreasing $\forall x \in R$
• C. strictly decreasing on $\left(-\infty, \frac{1}{\sqrt{3}}\right)$ and strictly increasing on $\left(\frac{1}{\sqrt{3}}, \infty\right)$
• D. strictly increasing on $\left(-\infty, \frac{1}{\sqrt{3}}\right)$ and strictly decreasing on $\left(\frac{1}{\sqrt{3}}, \infty\right)$

Answer 1

Answer: (A)

Assume $f ( x )= x ^3+ x$ so that $g ( x )= x ^3 ; h ( x )= x$ and $F ( x )= x ^3-\frac{\sqrt{3} x ^2}{2}+ x$ $F ^{\prime}( x )=3 x ^2-\sqrt{3} x +1$
$D =3-12<0$
$F ^{\prime}( x )>0 \forall x \in R \Rightarrow F$ is increasing $\Rightarrow$ (A)