Question

The sum of infintie terms of a decreasing G.P 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 $f^1\left(0\right)$, common ratio of the G.P is

• A. $1/3$
• B. $2/3$
• C. $4/3$
• D. $-2/3$

Answer 1

The correct option is B $2/3$

$f\left(x\right)=x^3+3x-9$

$f^{\prime }\left(x\right)=3x^2+3>0$
$f\left(x\right)\uparrow$

Greatest value =$f\left(3\right)$

$=27+9-9$

$=27$

$\frac{A}{1-R}=27$ ... (1)

$A-AR=f^{\prime }\left(0\right)=3$
$A(1-R)=3$ ... (2)

$\left(1\right)\times \left(2\right)$

$A^2=27\times 3=81$

$A=9$

$ \dfrac{9}{1-R}=27 \Rightarrow l-R=\dfrac{1}{3}$

$R=\frac{2}{3}$