In how many points the graph of `f(x)=x^3+2x^2+3x+4` meets the `x axis` ?

Explanation

To determine how many points the graph of the function \( f(x) = x^3 + 2x^2 + 3x + 4 \) meets the x-axis, we need to find the roots of the equation \( f(x) = 0 \). Here is a step-by-step solution:

### Step 1: Set the function equal to zero
We start by setting the function equal to zero to find the roots:
\[ x^3 + 2x^2 + 3x + 4 = 0 \]

### Step 2: Differentiate the function
Next, we differentiate the function to analyze its behavior:
\[ f'(x) = \frac{d}{dx}(x^3 + 2x^2 + 3x + 4) = 3x^2 + 4x + 3 \]

### Step 3: Analyze the derivative
To determine the nature of the function, we need to analyze the derivative \( f'(x) = 3x^2 + 4x + 3 \). We will find the discriminant of this quadratic function to see if it has any real roots:
\[ D = b^2 - 4ac = (4)^2 - 4(3)(3) = 16 - 36 = -20 \]

### Step 4: Determine the nature of the roots
Since the discriminant \( D \) is negative, the quadratic \( f'(x) \) has no real roots. This implies that \( f'(x) \) does not change sign and is always positive because the coefficient of \( x^2 \) (which is 3) is positive.

### Step 5: Conclude about the function's behavior
Since \( f'(x) > 0 \) for all \( x \), the function \( f(x) \) is always increasing. An always increasing function can meet the x-axis at most once.

### Step 6: Final conclusion
Thus, the graph of \( f(x) = x^3 + 2x^2 + 3x + 4 \) meets the x-axis at exactly one point.

### Summary
The graph of the function meets the x-axis at **one point**.

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