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

Category: QuestionsWhat is the nature of the function f(x) = 2/3(x^3) – 6x^2 + 20x – 5?
Editor">Editor Staff asked 11 months ago

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
 
This question was addressed to me during an online interview.
 
Asked question is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

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.