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What will be the range of the function f(x) = 2x^3 – 9x^2 – 24x + 5 which increases with x?(a) x > 4(b) x > 4 or x < -1(c) x < -1(d) Can’t be determinedI have been asked this question by my college director while I was bunking the class.This interesting question is from Calculus Application topic in chapter Application of Calculus of Mathematics – Class 12

Answer»

The correct option is (B) x > 4 or x < -1

Explanation: Since F(x) = 2x^3 – 9x^2 – 24x + 5

Therefore, f’(x) = 6x^2 – 18x + 24

= 6(x – 4)(x + 1)

If x > 4, then, x – 4 > 0 and x + 1 > 0

Thus, (x – 4)(x + 1) > 0 i.e., f’(x) > 0, when x > 4

Again, if x < -1, then, x – 4 < 0 and x + 1 < 0

So, from here,

(x – 4)(x + 1) > 0 i.e., f’(x) > 0, when x < -1

Hence, f’(x) > 0, when x > 4 Or x < -1

Therefore, f(x) increases with x when, x > 4 or x < -1



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