Functions

NOTE All integers are critical point for greatest integer
function.
Case I When x I
f (x) =
Case II When x I
If 0 < x < 1, then [x] = 0
and
Next, if

[x] = 1 and
Therefore, f (x) = if 1
Therefore, f (x) = 0, if 0
This shows that f (x) is continuous at x = 1.
Therefore, f (x) is discontinuous in
many other points.
Therefore, (b) is the answer.

Range
So option is correct.
f o g(x)
Range of g(x) Domain of f(x) = R
Common
Range of fog(x) = Range of f(x) when input

So option (B) is correct


So option is correct
g o f(x)
Range of f(x)
g o f(x)

(not possible)

So option is not correct.

There are only even numbers in Any subset of elements of will have at least odd number.

(A) is differentiable at every
(B) If , then is differentiable at every
(D) The derivative is equal to

(P) Minimum value of such that is continuous on
Let . We want to be continuous, meaning should not cross an integer on .
So, we need for a constant integer .
Step 1: Compute min and max of on

-

-
So . We want to lie inside an open interval of width 1, implying the range of must be less than 1: However, may not be sufficient, as the floor function could still jump.
Let’s try : Thus, for all , and the function is continuous.

So minimum is .


--- (Q) Minimum such that is increasing

Let’s check when : Since , is increasing if: Solve: So .


--- (R) Smallest such that is a local minimum for For local minima at , , and the sign of the derivative must change from negative to positive.
We analyze the multiplicity of the root at :
- : even implies a local min if the coefficient of other terms is>0.
Try (smallest even>5): This works, so .


--- (S) Check when not differentiable: modulus functions are not differentiable at . So the points where and are potential non-differentiable points for to .
So possible points:

-

-

We should check now if these all generate distinct results in , but if they do, then the total is 10.
Let . These are clearly continuous, but can affect differentiatibility.

Total: = (S) - (3)
--- Final Mapping:
- (P) → (5)

- (Q) → (1)

- (R) → (4)

- (S) → (3)
Match with options ⇒ Option (B)
Final Answer: