Statement I If `f(x)` is a continuous function defined from `R` to `Q` and `f(5)=3`, then differential coefficient of `f(x) w.r.t.x` will be `0`
Statement II Differentation of constant functions is always zero.

Options

A. Both statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I
C. Statement I is correct but Statement II is incorrect
D. Statement II is correct but Statement I is incorrect.

Answer

To solve the given question, we need to analyze both statements and determine their validity.

### Step 1: Analyze Statement I
Statement I states: "If \( f(x) \) is a continuous function defined from \( \mathbb{R} \) to \( \mathbb{Q} \) and \( f(5) = 3 \), then the differential coefficient of \( f(x) \) with respect to \( x \) will be \( 0 \)."

- A continuous function that takes a constant value (in this case, \( f(5) = 3 \)) means that the function does not change as \( x \) varies.
- Since \( f(x) \) is continuous and \( f(5) = 3 \), we can conclude that \( f(x) \) must be equal to \( 3 \) for all \( x \) in \( \mathbb{R} \) (because a continuous function that takes one value at one point must take that value everywhere if it is also a constant function).
- Therefore, \( f(x) = 3 \) for all \( x \).

### Step 2: Differentiate \( f(x) \)
- Now, we differentiate \( f(x) = 3 \) with respect to \( x \):
\[ \frac{df}{dx} = 0 \]
- This confirms that the differential coefficient of \( f(x) \) with respect to \( x \) is indeed \( 0 \).

### Conclusion for Statement I
Thus, Statement I is **true**.

### Step 3: Analyze Statement II
Statement II states: "Differentiation of constant functions is always zero."

- A constant function is defined as a function that does not change its value regardless of the input.
- The derivative of a constant function is defined as the rate of change of the function with respect to its variable. Since a constant function does not change, its rate of change is \( 0 \).

### Conclusion for Statement II
Thus, Statement II is also **true**.

### Final Conclusion
Both statements are true, and Statement II provides a correct explanation for Statement I. Therefore, the answer is:

**Option A: Both Statement I and Statement II are correct, and Statement II is the correct explanation of Statement I.**

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