Any tangent to the curve `y=2x^(5)+4x^(3)+7x+9`

Options

A. is parallel to x-axis
B. is parallel to y-axis
C. makes an acute angle with the x-axis
D. makes an obtuse angle with x-axis

Answer

To find any tangent to the curve \( y = 2x^5 + 4x^3 + 7x + 9 \), we need to follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( y \) with respect to \( x \) to determine the slope of the tangent line at any point on the curve.

\[ \frac{dy}{dx} = \frac{d}{dx}(2x^5 + 4x^3 + 7x + 9) \]

Using the power rule of differentiation:

\[ \frac{dy}{dx} = 2 \cdot 5x^{5-1} + 4 \cdot 3x^{3-1} + 7 \cdot 1 \]

This simplifies to:

\[ \frac{dy}{dx} = 10x^4 + 12x^2 + 7 \]

### Step 2: Analyze the slope
Next, we analyze the expression for the slope \( m = 10x^4 + 12x^2 + 7 \).

1. **Check if the slope can be zero**:
Since \( 10x^4 \) and \( 12x^2 \) are both non-negative for all real \( x \), and the constant term \( 7 \) is positive, we can conclude that:

\[ m = 10x^4 + 12x^2 + 7 > 0 \quad \text{for all } x \]

This means the slope of the tangent line is always positive.

### Step 3: Evaluate the options
Now we can evaluate the options given in the question:

1. **Option 1: Parallel to x-axis**: For a tangent to be parallel to the x-axis, the slope \( m \) must be \( 0 \). Since \( m > 0 \), this option is incorrect.

2. **Option 2: Parallel to y-axis**: For a tangent to be parallel to the y-axis, the slope \( m \) must be undefined. Since \( m \) is defined and positive, this option is also incorrect.

3. **Option 3: Acute angle with x-axis**: A tangent makes an acute angle with the x-axis if the slope \( m > 0 \). Since we have established that \( m > 0 \), this option is correct.

4. **Option 4: Obtuse angle with x-axis**: A tangent makes an obtuse angle with the x-axis if the slope \( m < 0 \). Since \( m > 0 \), this option is incorrect.

### Conclusion
The correct option is that the tangent to the curve makes an acute angle with the x-axis.

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