Question

suppose the total cost c(x) (in dollars) to manufacture a quantity x of weed killer (in hundreds of liters) is given by the function c(x)=x³ - 4x² + 8x + 40, where x > 0. a) where is c(x) decreasing? b) where is c(x) increasing? a) select the correct choice below and fill in any answer boxes within your choice. a. the function is decreasing on the open interval(s) □. (type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never decreasing. b) select the correct choice below and fill in any answer boxes within your choice. a. the function is increasing on the open interval(s) □. (type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never increasing.

Explanation:

Step1: Find the derivative of \(C(x)\)

Given \(C(x)=x^{3}-4x^{2}+8x + 40\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(C^\prime(x)=3x^{2}-8x + 8\).

Step2: Find the critical points

Set \(C^\prime(x) = 0\), so \(3x^{2}-8x + 8=0\). The discriminant \(\Delta=b^{2}-4ac=(-8)^{2}-4\times3\times8=64 - 96=-32<0\).

Step3: Analyze the sign of \(C^\prime(x)\)

Since the coefficient of \(x^{2}\) in \(C^\prime(x)=3x^{2}-8x + 8\) is \(a = 3>0\) and \(\Delta<0\), \(C^\prime(x)>0\) for all \(x>0\).

Answer:

a) \(B\). The function is never decreasing.
b) \(A\). The function is increasing on the open interval \((0,\infty)\)