Question

If ƒ is an even function such that $\frac{1}{2}\left( 1 - \sqrt{1 + 4\log_{2}x} \right)f(x) = 2^{x}$ has some finite non-zero value, then –

• A. ƒ is continuous and derivable at x = 0
• B. ƒ is continuous but not derivable at x = 0
• C. ƒ may be discontinuous at x = 0
• D. None of these

Answer 1

Answer: (B)

ƒ is continuous but not derivable at x = 0

Answer 2

Let ƒ′(0⁺) =$\frac { f ( \mathrm {~h} ) - f ( 0 ) } { \mathrm { h } }$ = k (say)

∴ƒ′(0^–) = $\frac { f ( 0 ) - f ( 0 - \mathrm { h } ) } { \mathrm { h } }$

= $\frac { f ( 0 ) - f ( \mathrm {~h} ) } { \mathrm { h } }$ = – k.

Q ƒ′(0⁺) ≠ ƒ′(0^–), but both are finite,

so ƒ(x) is continuous at x = 0 but not differentiable at x = 0.

Hence (2) is the correct answer.