Question

The function f(x)=2 x³ -33 x²+168 x-3 has two critical numbers. The smaller one is x= __ and the larger one is x=___

Answer 1

The critical numbers of the function f(x) = 2x³ - 33x² + 168x - 3 are found by setting its derivative equal to zero and solving the resulting quadratic equation. The smaller critical number is x = 4 and the larger one is x = 7.

Step-by-step explanation:

To find the critical numbers of the function f(x) = 2x³ - 33x² + 168x - 3, we need to calculate its derivative and then solve for x when the derivative is equal to zero. The derivative is f'(x) = 6x² - 66x + 168. Setting this equal to zero gives us a quadratic equation:

6x² - 66x + 168 = 0

To solve the quadratic equation, we divide through by 6 to simplify it to x² - 11x + 28 = 0, which can be factored to (x-4)(x-7) = 0. This gives us the two critical numbers when solving for x:

• x = 4 (the smaller critical number)
• x = 7 (the larger critical number)

Therefore, the smaller critical number is x = 4 and the larger one is x = 7.