Let `f (x) = {{:((x ^(p))/(( sin x ) ^(q) )"," , 0 lt x le (pi)/(2) ), ( 0 "," , x =0):}, (p,q in R).` Then
Lagrange's mean value theorem is applicable to f(x) in closed interval `[0,x],`

Options

A. for all p,q
B. only when `p gt q`
C. only when `p lt q`
D. for no alue of p,q

Answer

To determine if Lagrange's Mean Value Theorem (LMVT) is applicable to the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} \frac{x^p}{(\sin x)^q} & \text{for } 0 < x \leq \frac{\pi}{2} \\ 0 & \text{for } x = 0 \end{cases} \]

we need to check the conditions of continuity and differentiability on the closed interval \([0, x]\).

### Step 1: Check Continuity at \( x = 0 \)

To apply LMVT, we first need to check if \( f(x) \) is continuous at \( x = 0 \). For continuity at a point, we need:

\[ \lim_{x \to 0} f(x) = f(0) \]

Since \( f(0) = 0 \), we need to compute:

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{x^p}{(\sin x)^q} \]

### Step 2: Evaluate the Limit

Using the fact that \(\sin x \approx x\) as \(x \to 0\), we can rewrite the limit:

\[ \lim_{x \to 0} \frac{x^p}{(\sin x)^q} = \lim_{x \to 0} \frac{x^p}{(x)^q} = \lim_{x \to 0} x^{p-q} \]

### Step 3: Analyze the Limit Based on \( p \) and \( q \)

Now, we analyze the limit \( \lim_{x \to 0} x^{p-q} \):

1. **If \( p > q \)**:
- \( p - q > 0 \) implies \( \lim_{x \to 0} x^{p-q} = 0 \).

2. **If \( p < q \)**:
- \( p - q < 0 \) implies \( \lim_{x \to 0} x^{p-q} = \infty \).

3. **If \( p = q \)**:
- \( p - q = 0 \) implies \( \lim_{x \to 0} x^{p-q} = 1 \).

### Step 4: Conclusion on Continuity

For \( f(x) \) to be continuous at \( x = 0 \), we need:

\[ \lim_{x \to 0} f(x) = f(0) = 0 \]

This only holds true when \( p > q \).

### Step 5: Check Differentiability

Since \( f(x) \) is differentiable for \( 0 < x \leq \frac{\pi}{2} \) (as it is a quotient of differentiable functions), we only need to ensure that it is continuous at \( x = 0 \) for LMVT to apply.

### Final Conclusion

Thus, Lagrange's Mean Value Theorem is applicable to \( f(x) \) in the closed interval \([0, x]\) if and only if \( p > q \).

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