What is the nature of the function f(x) = 2/3(x^3) – 6x^2 + 20x – 5?

(a) Possess only minimum value

(b) Possess only maximum value

(c) Does not possess a maximum or minimum value

(d) Datainadequate

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Asked question is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12

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Right choice is (c) Does not possess a maximum or minimum value

Easiest explanation: We have, f(x) = 2/3(x^3) – 6x^2 + 20x – 5 ……….(1)

Differentiating both side of (1) with respect to x, we get,

f’(x) = 2x^2 – 12x + 20

Now, for a maximum and minimum value of f(x) we have,

f’(x) = 0

Or 2x^2 – 12x + 20 = 0

Or x^2 – 6x + 10= 0

So, x = [6 ± √(36 – 4*10)]/2

x = (6 ± √-4)/2, which is imaginary.

Hence, f’(x) does not vanishes at any point of x.

Thus, f(x) does not possess a maximum or minimum value.