A value of $$c$$ for which conclusion of Mean Value Theorem holds for the function $$f\left( x \right) = {\log _e}x$$ on the interval $$\left[ {1,3} \right]$$ is
A
$${\log _3}e$$
B
$${\log _e}3$$
C
$$2\,\,{\log _3}e$$
D
$${1 \over 2}{\log _3}e$$
Explanation
Using Lagrange's Mean Value Theorem
Let $$f(x)$$ be a function defined on $$\left[ {a,b} \right]$$
then, $$f'\left( c \right) = {{f\left( b \right) - f\left( a \right)} \over {b - a}}\,\,\,\,\,\,\,\,\,\,\,\,....\left( i \right)$$
$$c\,\, \in \left[ {a,b} \right]$$
$$\therefore$$ Given $$f\left( x \right) = {\log _e}x$$
$$\therefore$$ $$f'\left( x \right) = {1 \over x}$$
Hence, $$f(x)$$ is increasing when $$n \in \left( { - {\pi \over 2},{\pi \over 4}} \right)$$
3
AIEEE 2006
MCQ (Single Correct Answer)
A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$x$$. The maximum area enclosed by the park is
A
$${3 \over 2}{x^2}$$
B
$$\sqrt {{{{x^3}} \over 8}} $$
C
$${1 \over 2}{x^2}$$
D
$$\pi {x^2}$$
Explanation
Area $$ = {1 \over 2}{x^2}\,\sin \,\theta $$
Maximum value of $$\sin \theta $$ is $$1$$ at $$\theta = {\pi \over 2}$$
$${A_{\max }} = {1 \over 2}{x^2}$$
4
AIEEE 2006
MCQ (Single Correct Answer)
The function $$f\left( x \right) = {x \over 2} + {2 \over x}$$ has a local minimum at