Which graph represents the functionf(x)=x4+8x3+6x2−40x+25?

To identify the correct graph of the function \( f(x) = x^4 + 8x^3 + 6x^2 - 40x + 25 \), we can analyze key features such as the end behavior, critical points, and overall shape of the function.

### Step 1: **End Behavior**
Since the highest degree of the polynomial is \( x^4 \), the end behavior of the graph will follow the pattern:
- As \( x \to \infty \), \( f(x) \to \infty \)
- As \( x \to -\infty \), \( f(x) \to \infty \)

This is because even-degree polynomials (like \( x^4 \)) have both ends of the graph pointing in the same direction (upwards, in this case).

### Step 2: **Critical Points and Turning Points**
To find the critical points, we would need to compute the first derivative \( f'(x) \) and solve for points where \( f'(x) = 0 \).

The function's derivative is:
\[
f'(x) = 4x^3 + 24x^2 + 12x - 40
\]
Setting \( f'(x) = 0 \) and solving this cubic equation will give us the critical points, which correspond to local maxima, minima, or inflection points.

### Step 3: **Intercepts**
- **Y-intercept**: To find the y-intercept, set \( x = 0 \):
  \[
  f(0) = 0^4 + 8(0)^3 + 6(0)^2 - 40(0) + 25 = 25
  \]
  Thus, the y-intercept is \( (0, 25) \).
- **X-intercepts**: Solving \( f(x) = 0 \) for \( x \) involves finding the roots of the quartic polynomial, which may require numerical methods or graphing tools.

### Step 4: **Shape of the Graph**
Given that the function is a degree-4 polynomial with positive leading coefficient, it will have a "W"-like shape with possible multiple turning points (local minima and maxima).

Based on this analysis, the graph of \( f(x) = x^4 + 8x^3 + 6x^2 - 40x + 25 \) should:
- Rise to \( \infty \) as \( x \to \infty \) and \( x \to -\infty \)
- Have turning points (possibly 3 local maxima/minima)
- Intersect the y-axis at \( (0, 25) \)

### Next Steps:
You can verify the specific graph by plotting or checking each option if provided. 

Would you like help visualizing this or solving further?

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Here are 5 relative questions for practice:
1. What are the critical points of \( f(x) \)?
2. How many real roots can a quartic function have?
3. How does the leading coefficient affect the end behavior of polynomials?
4. How do you determine if a critical point is a local maximum or minimum?
5. What role do inflection points play in the graph's shape?

**Tip:** For polynomials, the number of possible turning points is at most the degree minus one.

Which graph represents the function f(x)=x^4+8x^3+6x^2−40x+25?

Explore how to analyze and graph the quartic polynomial f(x) = x^4 + 8x^3 + 6x^2 - 40x + 25. This solution discusses the end behavior, critical points, intercepts, and turning points of the polynomial. Ideal for high school students in Grades 10-12.

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