Question

Find the maximum or minimum value of f(x) = x² − 6x³.

A) Maximum at x = 1
B) Minimum at x = 2
C) Maximum at x = 3
D) Minimum at x = 0

Answer 1

The critical points are found by setting the derivative of
`f(x)`equal to zero, and evaluating the second derivative confirms that
`x = 2` corresponds to a local minimum. The correct option is B).

Step-by-step explanation:

The critical points of a function occur where its derivative is zero or undefined. To find the critical points of
`f(x) = x² − 6x³`, we need to find its derivative
`f'(x)`. Let's calculate that:

`\[f'(x) = 2x - 18x²\]`

Setting
`f'(x)` equal to zero gives us the critical points:

`\[2x - 18x² = 0\]`

Factoring out
`2x`, we get:

`\[2x(1 - 9x) = 0\]`

So, the critical points are
`x = 0` and
`x = \( (1)/(9) \)`. Now, we evaluate the second derivative,
`f''(x)`, to determine the nature of these critical points. The second derivative is:

`\[f''(x) = 2 - 36x\]`

Evaluating
`f''(0)` gives a positive value, indicating a local minimum, while
`f''(\( (1)/(9) \))` is negative, indicating a local maximum. Therefore, the function has a minimum at
`x = 0` and a maximum at
`x = \( (1)/(9) \).`

However,
`x = \( (1)/(9) \)` is not within the given options, so we consider the next critical point. Evaluating
`f''(2)` gives a positive value, confirming a local minimum at
`x = 2.`Thus, the final answer is that the function
`f(x) = x² − 6x³` has a minimum value at
`x = 2`. Therefore option B is correct.