Question

exercise 3.3.5
use technology to find the maximum and minimum values on the interval $-1,4$ of the function $f(x)=-0.2(x - 2)^3(x + 1)^2(x - 4)$.
minimum: $(x,y)=$
maximum: $(x,y)=$

Explanation:

Step1: Use graphing technology

Input the function \( f(x)=-0.2(x - 2)^{3}(x + 1)^{2}(x - 4) \) into a graphing calculator or software (such as Desmos).

Step2: Analyze the graph on the interval \([-1,4]\)

• Look for the lowest - point (minimum) and the highest - point (maximum) on the graph within the interval \([-1,4]\).

When we graph the function \(y =-0.2(x - 2)^{3}(x + 1)^{2}(x - 4)\) on the interval \([-1,4]\):

• For the minimum:

We observe that the minimum occurs at the point \((-1,0)\). Substitute \(x=-1\) into the function \(f(-1)=-0.2(-1 - 2)^{3}(-1 + 1)^{2}(-1 - 4)=0\) (since \((-1 + 1)^{2}=0\)).

• For the maximum:

The maximum occurs at the point \((2,0)\). Substitute \(x = 2\) into the function \(f(2)=-0.2(2 - 2)^{3}(2 + 1)^{2}(2 - 4)=0\) (since \((2 - 2)^{3}=0\)).

Answer:

Minimum: \((x,y)=(-1,0)\)
Maximum: \((x,y)=(2,0)\)