Analyzing the Extremes of the Cubic Function f(x) x3 - 6x2 24x - 4

In this article, we will discuss the maximum and minimum values of the cubic function f(x) x3 - 6x2 24x - 4. To understand whether this function has any critical points, we need to analyze its derivative and the roots of the resulting quadratic equation.

Derivation and Analysis of Extremes

The function given is f(x) x3 - 6x2 24x - 4. To find the critical points, we take the first derivative of the function. The first derivative is:

g(x) f'(x) 3x2 - 12x 24

Quadratic Equation and Roots

The quadratic equation derived from the first derivative is:

3x2 - 12x 24 0

Using the quadratic formula, x [-b ± √(b2 - 4ac)] / 2a, where a 3, b -12, and c 24, we can solve for the roots:

x [12 ± √((-12)2 - 4(3)(24))] / (2 * 3)

x [12 ± √(144 - 288)] / 6

x [12 ± √(-144)] / 6

Since the discriminant (144 - 288 -144) is negative, the roots of the quadratic equation are not real. Therefore, the first derivative g(x) 3x2 - 12x 24 does not have any real roots, which means the function f(x) x3 - 6x2 24x - 4 does not have any critical points. Consequently, the function does not have local maxima or minima.

No Extreme Values for the Function

The fact that the derivative does not have real roots indicates that the function f(x) x3 - 6x2 24x - 4 is a strictly increasing function. This is evident from the fact that the quadratic expression 3x2 - 12x 24 is always positive, as the discriminant is negative and the leading coefficient (3) is positive.

Since the function is strictly increasing, it does not have any maximum or minimum value. As x approaches positive infinity, also approaches positive infinity. Similarly, as x approaches negative infinity, f(x) approaches negative infinity. Therefore, there are no finite maximum or minimum values for this function over the set of real numbers.

Conclusion

The cubic function f(x) x3 - 6x2 24x - 4 does not have any extrema. The first derivative of the function does not have real roots, which indicates that the function is monotonically increasing. As a result, the function has no local maximum or minimum values, and it extends to positive and negative infinity as x goes to positive and negative infinity, respectively.

In summary, the key points to remember about the function f(x) x3 - 6x2 24x - 4 are:

The function does not have any critical points. The function is strictly increasing. The function has no local maximum or minimum values. The function has no finite bounds; it approaches ±∞ as x approaches ±∞.

Keywords

cubic function, maximum and minimum values, real roots