Identify the minimum value of the function *y* = 2*x*2 + 4*x*.

To find the minimum value of the function \( y = 2x^2 + 4x \), we can follow these steps:

### 1. Find the derivative of the function
First, we differentiate the function \( y = 2x^2 + 4x \) with respect to \( x \):

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

\[
\frac{dy}{dx} = 4x + 4
\]

### 2. Set the derivative equal to zero
To find the critical points, set the derivative equal to zero:

\[
4x + 4 = 0
\]

### 3. Solve for \( x \)
Solve the equation to find the value of \( x \):

\[
4x = -4
\]
\[
x = -1
\]

### 4. Determine if this point is a minimum
To confirm that this point is a minimum, we can check the second derivative. The second derivative of \( y \) is:

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}(4x + 4) = 4
\]

Since the second derivative is positive (\( 4 > 0 \)), the function is concave up, and the critical point \( x = -1 \) is indeed a minimum.

### 5. Find the minimum value of the function
Substitute \( x = -1 \) back into the original function to find the minimum value of \( y \):

\[
y = 2(-1)^2 + 4(-1) 
\]
\[
y = 2(1) - 4 
\]
\[
y = 2 - 4 
\]
\[
y = -2
\]

### Conclusion
The minimum value of the function \( y = 2x^2 + 4x \) is \( y = -2 \), which occurs at \( x = -1 \).

Would you like any more details or have any questions?

---

Here are 5 related questions to explore further:
1. How would the minimum value change if the coefficient of \( x^2 \) were different?
2. What happens to the minimum value if a constant is added to the function, say \( y = 2x^2 + 4x + 3 \)?
3. Can you determine the vertex of the parabola represented by \( y = 2x^2 + 4x \) without using calculus?
4. How would the function change if we instead maximized the function?
5. How do you interpret the concavity of the function based on the second derivative?

**Tip:** Always verify the nature of critical points by using the second derivative test; this helps confirm if the point is a minimum, maximum, or a saddle point.

Identify the minimum value of the function y = 2x^2 + 4x.

Discover how to find the minimum value of the quadratic function y = 2x^2 + 4x using calculus. This guide walks through key concepts like derivatives, the second derivative test, and optimization techniques, perfect for students in Grades 10-12.

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