Question

Find the values of x, such that f(x) is increasing function

f(x) = 2x³ – 15x – 144x – 7

Solution:

Given: f(x) = 2x³ – 15x² – 144 – 7

∴ f′(x) = 6x² – 30x – 144

Now, f′(x) > 0, as f(x) is increasing

∴ 6x² – 30x – 144 = 0

∴ x² – 5x – 24 = 0

∴ (x – 8)(x + 3) > 0

Case (I) x – 8 > 0 and x + 3 > 0

x > 8 and x > –3

∴ x > `square`

Case (II) x – 8 < 0 and x + 3 < 0

x < 8 and x < –3

∴ x < `square`

∴ f(x) = 2x³ – 15x² – 144 – 7 is increasing if and only if x ∈ (–∞, `square`) or x ∈ (`square`, ∞)

Answer 1

Given: f(x) = 2x³ – 15x² – 144 – 7

∴ f′(x) = 6x² – 30x – 144

Now, f′(x) > 0, as f(x) is increasing

∴ 6x² – 30x – 144 = 0

∴ x² – 5x – 24 = 0

∴ (x – 8)(x + 3) > 0

Case (I) x – 8 > 0 and x + 3 > 0

x > 8 and x > –3

∴ x > \[\boxed{8}\]

Case (II) x – 8 < 0 and x + 3 < 0

x < 8 and x < –3

∴ x < \[\boxed{–3}\]

∴ f(x) = 2x³ – 15x² – 144 – 7 is increasing if and only if x ∈ (–∞, \[\boxed{–3}\]) or x ∈ (\[\boxed{8}\], ∞)