Question

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

(a) Decreases

(b) Increases

(c) Cannot be determined for x < 1

(d) A constant function

This question was posed to me by my college director while I was bunking the class.

Query is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12

Answer 1

Correct answer 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 < 1, 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 < 1

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