Determine all of the relative extreme values of the function f (x) = 3x⁴ — 2x³ — 1  

The relative extreme value occurs at x = 1/3

The relative extreme values occur at x = 0 and 1/3

The relative extreme values occur at x = 1/2

There are no relative extreme values.

Answer:

The relative extreme values occur at x = ?1/2??.

Step-by-step explanation

f(x) = 3x⁴ - 2x³ - 1

The critical values occur at values of x where the first derivative of the function is 0.

i.e, f'(x) = 0

?dxd??(3x⁴ - 2x³ - 1) = 0

3* 4x³ - 2* 3x² - 0 = 0

12x³ - 6x² = 0

6x²(2x - 1) = 0

=> x = 0 and

(2x - 1) = 0, x = ?21??

Therefore x = 0 and x = ?21?? are the critical point.

Second derivative test:

For relative extreme, the second derivative must not equal 0.

Wherever the second derivative is 0, the point is an inflection point.

f"(x) = ?dxd??f'(x) = ?dxd??(12x³ - 6x²) = 12* 3x² - 6*2x = 36x² - 12x

---> At x = 0,

f"(0) = 36*0 - 12*0 = 0

The second derivative is 0, therefore at this value relative extreme will not occur.

----> At x = 1/2,

f(1/2) = 36*(??21??)² - 12 * ??21?? = 9 - 6 = 3

The second derivative is not equal to 0, therefore at this value of x, relative extreme will occur.

Answer:

The relative extreme values occur at x = ?1/2.