What will be nature of the f(x) = 10 – 9x + 6x^2 – x^3 for x > 3?

Category: QuestionsWhat will be nature of the f(x) = 10 – 9x + 6x^2 – x^3 for x > 3?
Editor">Editor Staff asked 11 months ago

What will be nature of the f(x) = 10 – 9x + 6x^2 – x^3 for x > 3?
 
(a) Decreases
 
(b) Increases
 
(c) Cannot be determined for x > 3
 
(d) A constant function
 
The question was posed to me during an online exam.
 
I’d like to ask this question from Calculus Application topic in portion Application of Calculus of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct option is (a) Decreases
 
For explanation: f(x) = 10 – 9x + 6x^2 – x^3
 
Thus, f’(x) =  – 9 + 12x –3 x^2
 
= -3(x^2 – 4x + 3)
 
Or f’(x) = -3(x – 1)(x – 3) ……….(1)
 
If x > 3, then x – 1 > 0 and x – 3 > 0
 
Hence, (x – 1)(x – 3) > 0
 
Thus, from (1) it readily follows that, f’(x) < 0, when x > 3
 
So, f(x) decreases for values of x > 3.