Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x∈R. If |f(x) f'(x) f'(x) f”(x)| = 0, for all x∈R, then the value of f(1) lies in the interval:

  1. (9, 12)
  2. (6, 9)
  3. (3, 6)
  4. (0, 3)

Answer (Detailed Solution Below)

Option 2 : (6, 9)

Detailed Solution

Download Solution PDF

Calculation:

GIven, f(x) f”(x) – f'(x)= 0

Let h(x) = f(x)/f'(x)

⇒ h'(x) = 0

⇒ h(x) = k

⇒ f(x)/f'(x) = k

⇒ f(x) = k f’(x)

Putting k = 0, we get

⇒ f(0) = k f'(0)

⇒ k = 1/2

Now, f(x) = \(\frac{1}{2}\) f'(x)

⇒ 2 = f'(x)/f(x)

Integrating on both sides, we get

⇒ ∫ 2dx = ∫ f'(x)/f(x) dx

⇒ 2x = ln |f(x)| + C

Now, f(0) = 1

⇒ C = 0

∴ 2x = ln |f(x)|

⇒ f(x) = ±e2x

As f(0) = 1

⇒ f(x) = e2x

∴ f(1) = e≈ 7.38

∴ The value of f(1) lies in the interval (6, 9).

The correct answer is Option 2.

More Applications of Derivatives Questions

More Differential Calculus Questions

Get Free Access Now
Hot Links: teen patti game - 3patti poker teen patti rules teen patti mastar teen patti gold apk