Let `f (x) = x ^(3//2) - sqrt ( x ^(2) + x ^(4)),` then

Options

A. L.H.D. at `x =0` exists but R.D.H. at `x =0` does not exist
B. f (x) is differentiable at x =0
C. R.H.D. at `x =0` exists but L.H.D. at `x =0` does not exists
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = x^{3/2} - \sqrt{x^2 + x^4} \) and determine the existence of the left-hand and right-hand derivatives at \( x = 0 \).

### Step 1: Calculate \( f(0) \)

First, we need to find the value of the function at \( x = 0 \):

\[ f(0) = 0^{3/2} - \sqrt{0^2 + 0^4} = 0 - 0 = 0 \]

### Step 2: Find the Right-Hand Derivative at \( x = 0 \)

The right-hand derivative at \( x = 0 \) is given by:

\[ f'_+(0) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} \]

Substituting \( f(0) = 0 \):

\[ f'_+(0) = \lim_{h \to 0^+} \frac{f(h)}{h} \]

Now, substituting \( f(h) \):

\[ f(h) = h^{3/2} - \sqrt{h^2 + h^4} \]

Thus,

\[ f'_+(0) = \lim_{h \to 0^+} \frac{h^{3/2} - \sqrt{h^2 + h^4}}{h} \]

### Step 3: Simplify the Expression

We can simplify the expression:

\[ f'_+(0) = \lim_{h \to 0^+} \left( h^{1/2} - \frac{\sqrt{h^2(1 + h^2)}}{h} \right) \]

This simplifies to:

\[ f'_+(0) = \lim_{h \to 0^+} \left( h^{1/2} - \sqrt{1 + h^2} \right) \]

### Step 4: Evaluate the Limit

As \( h \to 0 \):

\[ h^{1/2} \to 0 \quad \text{and} \quad \sqrt{1 + h^2} \to 1 \]

Thus,

\[ f'_+(0) = 0 - 1 = -1 \]

### Step 5: Find the Left-Hand Derivative at \( x = 0 \)

The left-hand derivative at \( x = 0 \) is given by:

\[ f'_-(0) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} \]

Substituting \( f(0) = 0 \):

\[ f'_-(0) = \lim_{h \to 0^-} \frac{f(h)}{h} \]

Now, substituting \( f(h) \):

\[ f(h) = (-h)^{3/2} - \sqrt{(-h)^2 + (-h)^4} \]

Since \( (-h)^{3/2} \) is not defined for negative \( h \) (as it results in a complex number), we cannot compute this limit.

### Step 6: Conclusion

Since the left-hand derivative does not exist and the right-hand derivative exists, we conclude:

- The right-hand derivative at \( x = 0 \) exists and is equal to \(-1\).
- The left-hand derivative at \( x = 0 \) does not exist.

Thus, the correct option is that the right-hand derivative at \( x = 0 \) exists, but the left-hand derivative does not.