Question

Determine all critical points for the following function. f (x )equals 2 x (4 minus x )cubedf(x)=2x(4−x)3 xequals=nothing ​(Use a comma to separate answers as​ needed.)

Answer 1

The critical points of the function

f(x) = 2x(4 - x)³

are x = (1, 4)

Explanation:

Given the function

f(x) = 2x(4 - x)³

To find the critical points, we differentiate f(x) to obtain f'(x), and then find the zeros of the resulting quadratic function.

Now, differentiate f(x).

f'(x) = 2(4 - x)³ + 2x(-1)×3(4 - x)²

= 2(4 - x)³ - 6x(4 - x)²

= 2(4 - x)²(4 - x- 3x)

= 2(4 - x)²(4 - 4x)

f'(x) = 8(4 - x)²(1 - x)

Now we set f'(x) = 0 and solve

8(4 - x)²(1 - x) = 0

=> 8(1 - x) = 0

=> 1 - x = 0

=> x = 1

Or

(4 - x)² = 0

x = 4 twice.

So x = (1, 4)

Therefore, the critical points are x = (1, 4)