Let be a function defined by `f(x)={{:(tanx/x", "x ne0),(1", "x=0):}`
Statement-1: x=0 is a point on minima of f
Statement-2: f'(0)=0

Options

A. Statement-1 is True, Statement-2 is True,Statement -2 is a correct explanation for Statement -7
B. Statement -1 True ,Statement -2 is True ,Stament -2 is not a correct explanation for Statement -!
C. Statement -1 is True Statement -2 is False
D. Statement -1 is Flase,Statement -2 is True

Explanation

To solve the problem, we will analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} \frac{\tan x}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \]

### Step 1: Check if \( x = 0 \) is a point of minima

To determine if \( x = 0 \) is a point of minima, we first need to find the derivative \( f'(x) \) for \( x \neq 0 \).

Using the quotient rule for differentiation:

\[ f'(x) = \frac{(x \cdot \sec^2 x) - (\tan x \cdot 1)}{x^2} \]

This simplifies to:

\[ f'(x) = \frac{x \sec^2 x - \tan x}{x^2} \]

### Step 2: Find critical points

To find critical points, we set \( f'(x) = 0 \):

\[ x \sec^2 x - \tan x = 0 \]

This implies:

\[ x \sec^2 x = \tan x \]

### Step 3: Evaluate the derivative at \( x = 0 \)

Next, we need to find \( f'(0) \). We can use the definition of the derivative:

\[ f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac{\frac{\tan x}{x} - 1}{x} \]

This limit is of the form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:

\[ f'(0) = \lim_{x \to 0} \frac{\sec^2 x - 0}{1} = \sec^2(0) = 1 \]

### Step 4: Analyze the sign of \( f'(x) \)

Now we analyze the sign of \( f'(x) \) around \( x = 0 \):

- For \( x > 0 \), \( f'(x) > 0 \) (since \( x \sec^2 x > \tan x \)).
- For \( x < 0 \), \( f'(x) < 0 \) (since \( x < \tan x \)).

### Step 5: Conclusion about minima

Since \( f'(x) \) changes from negative to positive as \( x \) passes through 0, we conclude that \( x = 0 \) is indeed a point of minima.

### Final Statements

- **Statement 1**: \( x = 0 \) is a point of minima of \( f \) is **true**.
- **Statement 2**: \( f'(0) = 0 \) is **false** since \( f'(0) = 1 \).

Thus, the correct answer is that Statement 1 is true, and Statement 2 is false.