What are the relative maximum and minimum points of the function f(x) = 2x^3 + x^2 + 11x?

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The relative maximum and minimum points of a function can be found by analyzing its critical points. A critical point occurs when the first derivative of the function is zero or undefined. Let’s go through the steps to find the critical points of the function f(x) = 2x³ + x² + 11x.

Step 1: Find the First Derivative

First, we need to calculate the first derivative of the function:

f'(x) = d/dx(2x³ + x² + 11x) = 6x² + 2x + 11

Next, we set the first derivative equal to zero to find the critical points:

6x² + 2x + 11 = 0

Step 2: Solve for Critical Points

Now, we can solve this quadratic equation. To determine the roots, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 6, b = 2, and c = 11:

b² - 4ac = 2² - 4(6)(11) = 4 - 264 = -260

Since the discriminant (b² – 4ac) is less than zero, this tells us that there are no real solutions to the equation. Therefore, there are no critical points where the first derivative is equal to zero.

Step 3: Determine the Nature of the Function

With no critical points, we assess the behavior of the function:

1. **Leading Term**: Since the highest degree term (2x³) has a positive coefficient and an odd degree, the function will diverge to +∞ as x approaches +∞ and -∞ as x approaches -∞.

2. **First Derivative Test**: Evaluating the first derivative at specific values helps in confirming whether the function is always increasing or decreasing. For all values of x, the first derivative f'(x) = 6x² + 2x + 11 will be positive because the quadratic expression’s discriminant is negative (no real roots). Therefore, f'(x) > 0 everywhere.

Conclusion

Since the function does not have any critical points and is always increasing, we conclude that there are no relative maximum or minimum points for the function f(x) = 2x³ + x² + 11x.

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