The curve `y=ax^(3)+bx^(2)+cx` is inclined at `45^(@)` to x-axis at `(0,0)` but it touches x-axis at `(1,0)`, then

Options

A. f'(1) = 0
B. f''(1) = 2
C. f'''(2) = 12
D. f(2) = 2

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the necessary equations.

### Step 1: Establish the function and conditions
The curve is given by the equation:
\[ y = ax^3 + bx^2 + cx \]

We know two key conditions:
1. The curve is inclined at \( 45^\circ \) to the x-axis at the point \( (0,0) \).
2. The curve touches the x-axis at the point \( (1,0) \).

### Step 2: Use the first condition
The slope of the curve at \( (0,0) \) is given by the derivative \( y' \) evaluated at \( x = 0 \). The slope of a line inclined at \( 45^\circ \) is \( 1 \).

First, we find the derivative:
\[ y' = \frac{dy}{dx} = 3ax^2 + 2bx + c \]

Now, we evaluate this at \( x = 0 \):
\[ y'(0) = 3a(0)^2 + 2b(0) + c = c \]

Setting this equal to \( 1 \) (the slope at \( 45^\circ \)):
\[ c = 1 \] (Equation 1)

### Step 3: Use the second condition
Since the curve touches the x-axis at \( (1,0) \), we have:
1. \( y(1) = 0 \)
2. The derivative \( y'(1) = 0 \) (because it touches the x-axis).

Substituting \( x = 1 \) into the original equation:
\[ y(1) = a(1)^3 + b(1)^2 + c(1) = a + b + c = 0 \]
Substituting \( c = 1 \) from Equation 1:
\[ a + b + 1 = 0 \]
Thus, we have:
\[ a + b = -1 \] (Equation 2)

Now, we find the derivative at \( x = 1 \):
\[ y'(1) = 3a(1)^2 + 2b(1) + c = 3a + 2b + c = 0 \]
Substituting \( c = 1 \):
\[ 3a + 2b + 1 = 0 \]
Thus, we have:
\[ 3a + 2b = -1 \] (Equation 3)

### Step 4: Solve the system of equations
Now we have two equations:
1. \( a + b = -1 \) (Equation 2)
2. \( 3a + 2b = -1 \) (Equation 3)

From Equation 2, we can express \( b \) in terms of \( a \):
\[ b = -1 - a \]

Substituting this into Equation 3:
\[ 3a + 2(-1 - a) = -1 \]
\[ 3a - 2 - 2a = -1 \]
\[ a - 2 = -1 \]
\[ a = 1 \]

Now substituting \( a = 1 \) back into Equation 2 to find \( b \):
\[ 1 + b = -1 \]
\[ b = -2 \]

### Step 5: Final values
We have:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 1 \)

### Step 6: Write the final equation of the curve
Substituting \( a \), \( b \), and \( c \) into the original equation:
\[ y = 1x^3 - 2x^2 + 1x \]
Thus, the equation of the curve is:
\[ y = x^3 - 2x^2 + x \]