Let `f (x)` be a continuous function (define for all x) which satisfies `f ^(3) (x)-5 f ^(2) (x)+ 10f (x) -12 ge 0, f ^(2) (x) + 3 ge 0 and f ^(2) (x) -5f(x)+ 6 le 0`
The equation of tangent that can be drawn from `(2,0)` on the curve `y = x^(2) f (sin x)` is :

Options

A. `y=24 (x+2)`
B. ` y= 12 (x+2)`
C. `y=24(x-2)`
D. `y=12 (x-2)`

Answer

To solve the problem step by step, we will analyze the given inequalities and find the equation of the tangent line from the point (2, 0) on the curve \( y = x^2 f(\sin x) \).

### Step 1: Analyze the inequalities

We have three inequalities involving the function \( f(x) \):

1. \( f^{(3)}(x) - 5 f^{(2)}(x) + 10 f(x) - 12 \geq 0 \)
2. \( f^{(2)}(x) + 3 \geq 0 \)
3. \( f^{(2)}(x) - 5f(x) + 6 \leq 0 \)

Let's denote \( f(x) = t \). Then we can rewrite the inequalities in terms of \( t \).

### Step 2: Rewrite the first inequality

The first inequality becomes:

\[ t^3 - 5t^2 + 10t - 12 \geq 0 \]

### Step 3: Factor the cubic polynomial

We can factor the cubic polynomial. Let's find the roots using the Rational Root Theorem or synthetic division.

After testing possible rational roots, we find that \( t = 2 \) is a root. We can factor it as follows:

\[ t^3 - 5t^2 + 10t - 12 = (t - 2)(t^2 - 3t + 6) \]

### Step 4: Analyze the quadratic

Next, we analyze the quadratic \( t^2 - 3t + 6 \). The discriminant \( D \) is:

\[ D = (-3)^2 - 4 \cdot 1 \cdot 6 = 9 - 24 = -15 \]

Since the discriminant is negative, \( t^2 - 3t + 6 > 0 \) for all \( t \). Therefore, the first inequality holds if \( t - 2 \geq 0 \) or \( t \geq 2 \).

### Step 5: Analyze the second inequality

The second inequality \( f^{(2)}(x) + 3 \geq 0 \) becomes:

\[ t^2 + 3 \geq 0 \]

This is always true for all real \( t \).

### Step 6: Analyze the third inequality

The third inequality \( f^{(2)}(x) - 5f(x) + 6 \leq 0 \) becomes:

\[ t^2 - 5t + 6 \leq 0 \]

Factoring gives:

\[ (t - 2)(t - 3) \leq 0 \]

### Step 7: Find the intervals for \( t \)

The solution to this inequality is:

\[ 2 \leq t \leq 3 \]

### Step 8: Combine results

From the first inequality, we found \( t \geq 2 \), and from the third inequality, we found \( 2 \leq t \leq 3 \). Thus, we conclude:

\[ t = f(x) = 3 \]

### Step 9: Substitute back into the curve equation

Now we substitute \( f(x) = 3 \) into the curve equation \( y = x^2 f(\sin x) \):

\[ y = x^2 \cdot 3 = 3x^2 \]

### Step 10: Find the derivative

To find the slope of the tangent line at the point (2, 0), we differentiate \( y \):

\[ \frac{dy}{dx} = 6x \]

At \( x = 2 \):

\[ \frac{dy}{dx} \bigg|_{x=2} = 6 \cdot 2 = 12 \]

### Step 11: Write the equation of the tangent line

Using the point-slope form of the line:

\[ y - y_1 = m(x - x_1) \]

Where \( (x_1, y_1) = (2, 0) \) and \( m = 12 \):

\[ y - 0 = 12(x - 2) \]

Simplifying gives:

\[ y = 12x - 24 \]

### Final Answer

Thus, the equation of the tangent line is:

\[ \boxed{y = 12x - 24} \]