Applications Of Derivatives Question 428
Question: The number of solutions of the equation $ 3\tan x+x^{3}=2in( 0,\frac{\pi }{4} ). $ is
Options:
A) 1
B) 2
C) 3
D) Infinite
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ f(x)=3\tan ,x+x^{3}-2 $ . Then $ f’(x)=3{{\sec }^{2}}x+3x^{2}>0 $ .
Hence, f(x) increases.
Also, $ f(0)=-2 $ and $ f( \frac{\pi }{4} )>0 $ .
So, by intermediate value theorem, $ f(c)=2 $ for some $ c\in ( 0,\frac{\pi }{4} ) $ .
Hence, $ f(x)=0 $ has only one root.
BETA
