Question

Let f(x)=x^α Inx, if x>0 and f(x)=0, if x=0. Then Rolle's theorem is applicable to f for x∈[0,1], if α has the value or values
A) 2
B) 1
C) 0
D) 1/2
​

Answer 1

Answer:

A , B , D

Step-by-step explanation:

For Rolle's Theorem to applicable function must be continuous on close interval [ 0 , 1] . Here only problem that can be is discontinuity at

x=0 . f(x) should go to 0 as we approach zero .

Let's calculate limit using L hopital rule

$\displaystyle{\lim_{x \to 0} \: \frac{ ln(x) }{ {x}^{ - \alpha } } }= \frac{1}{- \alpha {x}^{ - \alpha } } = \frac{ {x}^{ \alpha } }{ - \alpha }$

Now only for

$\alpha = 0 \\ \displaystyle{\lim_{x \to 0}f(x) = \frac{ - 1}{ 0} } \neq \: 0$

So we can't apply Rolle only for this value .