Consider functions f x = x , - 1 ≤ x ≤ 1 x , 1 < x ≤ 2 and g x = x , - 2 < x ≤ 1 x , 1 < x ≤ 2 (where · represents fractional part function). Then which of the following is/are incorrect?

Number of points of differentiability of f x is one

Consider a function f x = sgn x + sgn x + sgn x defined in the interval - 5 , 5 , where · and · represent the greatest integer and the fractional part functions respectively. Then

Number of points of discontinuity of the function f x is nine

If the range of the function f x is the set B , then n B = 6

Number of integral solution(s) of the equation f x = 3 , is one

Let a function f : - 2 , 2 → B be defined as f x = min x 3 , sin - 1 x 3 8 , cos - 1 sin x . Then

If L . M . V . T is applicable to y = f x in a , b , then b - a can be 2

Number of points of non-differentiability of f x is three

If L . M . V . T is applicable to y = f x in a , b , then b - a can be 3

If L . M . V . T is applicable to y = f x in a , b , then b - a can be 1

If f x = x 2 - 2 x and g x = min . { f ( t ) : - 2 ≤ t ≤ x , - 2 ≤ x ≤ 0 max . { f ( t ) : 0 ≤ t ≤ x , 0 ≤ x ≤ 3 . Then

g x is not differentiable at x = 2

If x y = e x - y , prove that d y d x = log x 1 + log x 2 .

Given: x y = e x - y . Applying logarithm on both sides. y log x = x - y ⇒ y + y log x = x y = x 1 + log x On differentiating both sides with respect to x , we get d y d x = 1 + log x · 1 - x 1 x 1 + log x 2 ⇒ d y d x = 1 + log x - 1 1 + log x 2 ⇒ d y d x = log x 1 + log x 2 Hence, proved.