Equation of normal drawn to the graph of the function defined as f(x) = `(sinx^2)/x` , `x !=0` and f(0) =0, at the origin.

Options

A. `x + y =0`
B. `x-y=0`
C. `y=0`
D. `x=0`

Answer

To find the equation of the normal drawn to the graph of the function \( f(x) = \frac{\sin(x^2)}{x} \) at the origin, we will follow these steps:

### Step 1: Find the derivative \( f'(x) \)

We need to find the derivative of the function \( f(x) \). We will use the quotient rule for differentiation, which states:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \]

Here, let \( u = \sin(x^2) \) and \( v = x \).

- First, we find \( u' \):
\[ u' = \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = 2x \cos(x^2) \]

- Next, we find \( v' \):
\[ v' = \frac{d}{dx}(x) = 1 \]

Now, applying the quotient rule:
\[ f'(x) = \frac{(2x \cos(x^2))(x) - (\sin(x^2))(1)}{x^2} = \frac{2x^2 \cos(x^2) - \sin(x^2)}{x^2} \]

### Step 2: Simplify \( f'(x) \)

We can simplify \( f'(x) \):
\[ f'(x) = 2 \cos(x^2) - \frac{\sin(x^2)}{x^2} \]

### Step 3: Find the slope of the tangent at the origin

To find the slope of the tangent line at the origin (0, 0), we need to evaluate \( f'(x) \) as \( x \) approaches 0:
\[ \text{slope of tangent} = \lim_{x \to 0} f'(x) = \lim_{x \to 0} \left( 2 \cos(x^2) - \frac{\sin(x^2)}{x^2} \right) \]

Using the fact that \( \cos(0) = 1 \) and \( \lim_{x \to 0} \frac{\sin(x^2)}{x^2} = 1 \):
\[ \text{slope of tangent} = 2 \cdot 1 - 1 = 1 \]

### Step 4: Find the slope of the normal

The slope of the normal line is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{1} = -1 \]

### Step 5: Write the equation of the normal line

The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line. Here, the point is the origin (0, 0) and the slope is -1:
\[ y - 0 = -1(x - 0) \]
This simplifies to:
\[ y = -x \]

### Final Answer

The equation of the normal drawn to the graph of the function at the origin is:
\[ y = -x \]