If `f:R to R ` given by `f(x)=x^(3)+px^(2)+qx+r,` is then find the condition for which `f(x)` is one-one.

Explanation

To determine the condition under which the function \( f(x) = x^3 + px^2 + qx + r \) is one-one, we need to analyze its derivative.

### Step-by-Step Solution:

1. **Find the Derivative**:
The first step is to find the derivative of the function \( f(x) \).
\[ f'(x) = \frac{d}{dx}(x^3 + px^2 + qx + r) = 3x^2 + 2px + q \]

2. **Condition for One-One Function**:
A function is one-one if its derivative does not change sign. This means that \( f'(x) \) should either be always positive or always negative. For a quadratic function \( ax^2 + bx + c \) to be always positive, its discriminant must be less than or equal to zero.

3. **Calculate the Discriminant**:
The discriminant \( D \) of the quadratic \( 3x^2 + 2px + q \) is given by:
\[ D = b^2 - 4ac = (2p)^2 - 4(3)(q) = 4p^2 - 12q \]

4. **Set the Discriminant Condition**:
For \( f'(x) \) to be non-negative (which means \( f(x) \) is one-one), we need:
\[ D \leq 0 \implies 4p^2 - 12q \leq 0 \]

5. **Simplify the Inequality**:
Dividing the entire inequality by 4 gives:
\[ p^2 - 3q \leq 0 \]
Rearranging this, we find:
\[ p^2 \leq 3q \]

### Conclusion:
The condition for the function \( f(x) = x^3 + px^2 + qx + r \) to be one-one is:
\[ p^2 \leq 3q \]