Problem 110 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that the cubic function $$ f(x)=a x^{3}+b x^{2}+c x+d \quad(a \neq 0) $$ has no relative extremum if and only if \(b^{2}-3 a c \leq 0\).

Step-by-Step Solution

Answer

A cubic function \(f(x) = ax^3 + bx^2 + cx + d\) with \(a \neq 0\) has no relative extremum if and only if \(b^2 - 3ac \leq 0\). We established this condition by analyzing the discriminant of the first derivative, \(f'(x) = 3ax^2 + 2bx + c\). If \(b^2 - 3ac \leq 0\), the derivative will not have real solutions, indicating no relative extrema in the cubic function.

Step 1: Finding First Derivative

To find the critical points of the function, we need to find its first derivative. Calculate the first derivative, \(f'(x)\), of the cubic function: $$ f(x) = ax^3 + bx^2 + cx + d $$ Applying the power rule, we get: $$ f'(x) = 3ax^2 + 2bx + c $$

Step 2: Finding Critical Points of the Function

To find the critical points of the function, set \(f'(x)\) equal to zero: $$ 3ax^2 + 2bx + c = 0 $$ We are looking for x-values that satisfy the above equation. This is a quadratic equation in the form of \(Ax^2 + Bx + C = 0\), where \(A = 3a, B = 2b\), and \(C = c\). To determine if the equation has real solutions, we will use the discriminant formula: $$ \Delta = B^2 - 4AC $$ Here, we want to analyze if this quadratic equation has real zeros, which will serve as the critical points for the function's extremum.

Step 3: Evaluating the Discriminant

Substitute the values of \(A, B\), and \(C\) into the discriminant formula: $$ \Delta = (2b)^2 - 4(3a)(c) $$ Simplify the discriminant: $$ \Delta = 4b^2 - 12ac $$ For a quadratic equation, there will be real solutions if \(\Delta > 0\), a repeated solution if \(\Delta = 0\), and no real solutions if \(\Delta < 0\). Since we are trying to prove the condition \(b^2 - 3ac \leq 0\) for the non-existence of relative extrema, let's rewrite the discriminant inequality in the form we need: $$ b^2 - 3ac \geq \frac{1}{3}\Delta $$ In other words, the cubic function will not have a relative extremum if \(b^2 - 3ac \leq 0\) because the derivative will not have real solutions under this condition.

Step 4: Conclusion

A cubic function \(f(x) = ax^3 + bx^2 + cx + d\) with \(a \neq 0\) has no relative extremum if and only if: $$ b^2 - 3ac \leq 0 $$ By analyzing the discriminant of the first derivative, we can deduce this condition.

Key Concepts

Critical Points in Calculus

In calculus, critical points are significant because they are the x-values where the slope of the function is zero or undefined; essentially, they're where the graph of the function has a horizontal tangent line or a cusp. These points are monumental in identifying the function's local maxima and minima—commonly known as