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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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.