Question

The sum of infinite terms of decreasing GP is equal to the greatest value of the function $f\left(x\right)=x^3+3x-9$ in the interval [-2, 3] and the difference between the first two terms is $f^{{}^{\prime }}\left(0\right)$. Then the common ratio of the GP is

• A. $-\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{2}{3}$
• D. $-\frac{4}{3}$

Answer 1

Answer: (C)

$\frac{2}{3}$

The function $f\left(x\right)=x^3+3x-9$ is an increasing function

$\because f^{{}^{\prime }}\left(x\right)=3x^2+3>0\forall [-2,3]$

$\therefore$ Maximum value $=27$

Thus, $\frac{a}{1-r}=27$ and $a-ar=3$

$\Rightarrow a=\frac{3}{1-r}$

$\therefore (\frac{1}{1-r}{)}^2=9$

$\Rightarrow \frac{1}{1-r}=\pm 3$

$\Rightarrow 1-r=\pm \frac{1}{3}$

$\Rightarrow r=\frac{2}{3}$

$r\ne \frac{4}{3}$ as decreasing GP.