closed interval method: problem 1\n(1 point)\nthe function ( f(x)=2 x^{3}-18 x^{2}+30 x - 11 ) has two…

closed interval method: problem 1\n(1 point)\nthe function ( f(x)=2 x^{3}-18 x^{2}+30 x - 11 ) has two critical values. the smaller one equals ( square ) and the larger one equals ( square )\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

closed interval method: problem 1\n(1 point)\nthe function ( f(x)=2 x^{3}-18 x^{2}+30 x - 11 ) has two critical values. the smaller one equals ( square ) and the larger one equals ( square )\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=2x^{3}-18x^{2}+30x - 11) is (f^{\prime}(x)=6x^{2}-36x + 30) (using the power rule ((x^{n})^\prime=nx^{n - 1})).

Step2: Set the derivative equal to zero

Set (f^{\prime}(x)=0), so (6x^{2}-36x + 30 = 0). Divide through by (6) to get (x^{2}-6x + 5=0).

Step3: Factor the quadratic equation

Factor (x^{2}-6x + 5=(x - 1)(x - 5)=0).

Step4: Solve for (x)

Using the zero - product property (x-1 = 0) gives (x = 1) and (x - 5=0) gives (x = 5).

Answer:

The smaller one equals (1) and the larger one equals (5).