Question

If ƒ(x) = ax³ + bx² + cx + d where a, b, c, d are real numbers and 3b²< c², is an increasing cubic function and

g(x) = aƒ′(x) + bƒ′′(x) + c², then –

• A. $\int_{a}^{x}{g(t)}$dt is a decreasing function
• B. $\int_{a}^{x}{g(t)}$dt is an increasing function
• C. $\int_{a}^{x}{g(t)}$ dt is a neither increasing nor decreasing function
• D. None of the above

Answer 1

Answer: (B)

$\int_{a}^{x}{g(t)}$dt is an increasing function

Answer 2

ƒ′(x) = 3ax² + 2bx + c > 0 (since ƒ(x) is increasing)

⇒ a > 0 and b² – 3ac < 0

⇒ a > 0 and b² < 3ac

also, g(x) = aƒ′(x) + bƒ′′(x) + c²

g(x) = 3a²x² + 2abx + ac + 6abx + 2b² + c²

g(x) = 3a²x² + 8abx + (2b² + c² + ac)

where D = 64a²b² – 4 · 3a² · (2b² + c² + ac)

= 4a² (16b² – 6b² – 3c² – 3ac)

= 4a² (10b² – 3c²– 3ac) < 4a² (10b² – 3c² – b²)

{as 3ac > b² ⇒ – 3ac < – b²}

= 4a² (9b² – 3c²)

= 12a² (3b² – c²) {given 3b² < c²}

∴ D > 0

⇒ g(x) < 0, ∀ x ∈ R

∴ $\int_{0}^{x}{g(t)}$ dt is an increasing function.

Hence (2) is the correct answer.