What will be the range of the function f(x) = 2x^3 – 9x^2 – 24x + 5 which decreases with x?

Category: QuestionsWhat will be the range of the function f(x) = 2x^3 – 9x^2 – 24x + 5 which decreases with x?
Editor">Editor Staff asked 11 months ago

What will be the range of the function f(x) = 2x^3 – 9x^2 – 24x + 5 which decreases with x?
 
(a) -1 < x < 4
 
(b) 1 < x < 4
 
(c) -1 ≤ x < 4
 
(d) -1 < x ≤ 4
 
The question was posed to me by my school principal while I was bunking the class.
 
My question is from Calculus Application topic in chapter 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 answer is (a) -1 < x < 4
 
To explain: Since f(x) = 2x^3 – 9x^2 – 24x + 5
 
Therefore, f’(x) = 6x^2 – 18x + 24
 
= 6(x – 4)(x + 1)
 
If -1 < x < 4, then x – 4 < 0 and x + 1 > 0
 
Thus, (x – 4)(x + 1) < 0 i.e.,
 
f’(x) < 0, when -1 < x < 4
 
Therefore, f(x) decreases with x, when, -1 < x < 4.