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
For maxima or minima,
⇒ 6(x + 1)(x − 2) = 0
⇒ x + 1 = 0 or x − 2 = 0
⇒ x = −1 or x = 2
Now,
At x = −1, we have
< 0
So, x = −1 is the point of local maximum.
At x = 2, we have
> 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).