Question

The function f(x) = 2x³ - 3x² - 12x + 4, has
(a) two points of local maximum (b) two points of local minimum
(c) one maximum and one minimum (d) no maximum no minimum

Answer 1

The given function is f(x) = 2x³ − 3x² − 12x + 4.

f(x) = 2x³ − 3x² − 12x + 4

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=6x^2-6x-12$

$\Rightarrow f\text{'}\left(x\right)=6\left(x^2-x-2\right)$

$\Rightarrow f\text{'}\left(x\right)=6\left(x+1\right)\left(x-2\right)$

For maxima or minima,

$f\text{'}\left(x\right)=0$

⇒ 6(x + 1)(x − 2) = 0

⇒ x + 1 = 0 or x − 2 = 0

⇒ x = −1 or x = 2

Now,

$f\text{'}\text{'}\left(x\right)=12x-6$

At x = −1, we have

$f\text{'}\text{'}\left(-1\right)=12\times \left(-1\right)-6=-12-6=-18$ < 0

So, x = −1 is the point of local maximum.

At x = 2, we have

$f\text{'}\text{'}\left(2\right)=12\times 2-6=24-6=18$ > 0

So, x = 2 is the point of local minimum.

Thus, the given function f(x) = 2x³ − 3x² − 12x + 4 has one point of local maximum and one point of local minimum.

Hence, the correct answer is option (c).