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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

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Solution


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'x=6x2-6x-12

f'x=6x2-x-2

f'x=6x+1x-2

For maxima or minima,

f'x=0

⇒ 6(x + 1)(x − 2) = 0

⇒ x + 1 = 0 or x − 2 = 0

⇒ x = −1 or x = 2

Now,

f''x=12x-6

At x = −1, we have

f''-1=12×-1-6=-12-6=-18 < 0

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

At x = 2, we have

f''2=12×2-6=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).

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