Applications Of Derivatives Question 154
Question: The interval in which the function $ x^{3} $ increases less rapidly than $ 6x^{2}+15x+5 $ , is
Options:
A) $ (-\infty ,,-1) $
B) (-5 , 1)
C) (-1 ,5)
D) (5 , $ \infty $ )
Show Answer
Answer:
Correct Answer: C
Solution:
The function $ f(x)=x^{3} $ increases for all x and the function $ g(x)=6x^{2}+15x+5 $ increases, if $ g’(x)>0\Rightarrow 12x+15>0\Rightarrow x>-\frac{5}{4} $ .
Thus $ f(x) $ and $ g(x) $ both increases for $ x>-\frac{5}{4} $ .
It is given that $ f(x) $ increases less rapidly than $ g(x) $ ,
Therefore the function $ \varphi (x)=f(x)-g(x) $ is decreasing function , which implies that $ \varphi ‘(x)<0 $
therefore $ 3x^{2}-12x-15<0\Rightarrow -1<x<5 $ .
BETA
