Question

If f(x) = x^alog x and f(0) = 0, then the value of a for which Rolle's theorem can be applied in [0, 1] is –

• A. – 2
• B. – 1
• C. 0
• D. $\frac{1}{2}$

Answer 1

Answer: (D)

$\frac{1}{2}$

Answer 2

If f(x) satisfy roll's theorem in [0, 1]

Ž f(x) must be continuous in [0, 1]

Ž f(0 +) = f(0)

Ž $\lim_{h \rightarrow 0}$h^a logh = 0

Ž $\lim _ { h \rightarrow 0 }$ $\frac{\log h}{h–\alpha}$ = 0

Ž $\lim_{h \rightarrow 0}$ $\frac{1/h}{–\alpha h^{–\alpha –1}}$ = 0

Ž $\lim_{h \rightarrow 0}–\frac{h^{\alpha}}{\alpha}$ = 0

for continuous at x = 0, a > 0

\ a = ½