Question

What will be nature of the f(x) = 10 – 9x + 6x^2 – x^3 for 1 < x < 3?

(a) Decreases

(b) Increases

(c) Cannot be determined for 1 < x < 3

(d) A constant function

I had been asked this question during a job interview.

This key question is from Calculus Application topic in chapter Application of Calculus of Mathematics – Class 12

Answer 1

Correct choice is (b) Increases

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 1 < 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 1 < x < 3

So, f(x) increases for values of 1 < x < 3.