Applications Of Derivatives Question 19
Question: The range of the function $ f(x)=2\sqrt{x-2}+\sqrt{4-x} $ is
Options:
A) $ ( \sqrt{2},\sqrt{10} ) $
B) $ [ \sqrt{2},\sqrt{10} ) $
C) $ ( \sqrt{2},\sqrt{10} ] $
D) $ [ \sqrt{2},\sqrt{10} ] $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Clearly, domain of the function is [2, 4]. Now, $ f’(x)=\frac{1}{\sqrt{x-2}}-\frac{1}{2\sqrt{4-x}} $
$ f’(x)=0 $ or $ \sqrt{x-2}=2\sqrt{4-x} $ or $ x-2=16-4x $ or $ x=\frac{18}{5} $ Now, $ f(2)=\sqrt{2},f( \frac{18}{5} )=2\sqrt{\frac{18}{5}-2}+\sqrt{4-\frac{18}{5}}=\sqrt{10}, $
$ f(4)=2\sqrt{2} $
Hence, range of the function is $ [\sqrt{2},\sqrt{10}] $ . Also, here $ x=(18/5) $ is the point of global maxima.
BETA
