An interesting function

Calculus Level 4

$f(x)$ is a non-negative function defined for $x\geq1$ such that the inequality $f '(x)\leq m \cdot f(x)$ holds everywhere in the domain for some positive real number $m$ . If $f(1)=0$ then find the value of $f(e)+f(e^2)$ .

The answer is 0.

1 solution

Soutik Banerjee
May 2, 2016

Here is my solution.

Could you explain your motivation to choose $g(x)$ as $e^{-mx} \cdot f(x)$ initially?

Pranshu Gaba - 5 years, 1 month ago

The form of the inequality in the problem kind of gives a hint. Exponential functions being used for their property of retaining form after differentiation is pretty common. In this particular problem the fact that $e^x$ is always positive comes in handy as well.

Soutik Banerjee - 5 years, 1 month ago

Yes, that makes sense, although it wasn't intuitive for me. I will keep this form in mind while solving problems in future.

Nonetheless, the solution is beautiful and elegant. :)

Pranshu Gaba - 5 years, 1 month ago

An interesting function

The answer is 0.

1 solution

0 pending reports