Prove that a function is strictly increasing

calculusfunctionssequences-and-series

I have the following function
$$
f(x)= x\left (1 + \displaystyle{\sum_{j=1}^M \ \frac{p}{1 + D_{j}\cdot x}} \right)- t.
$$

$x$ is the only variable (the others are positive constants). We have been asked to show two things:

  1. Function is positive for some large values of $x$.
  2. Function is strictly increasing for positive $x$.

For (1) I can just use a few values to prove it. For the second part, I'm thinking of taking the derivative of this function and proving that it will always be positive. I got the derivative and said that for very large values of $x$ (as $x\to\infty$), the derivative will converge to $1$. Thus it will be strictly increasing. Just hoping someone can help me confirm this.

Best Answer

I got the derivative and said that for very large values of $x$ (as $x\to\infty$), the derivative will converge to $1$

All that this proves is that $f(x)$ is increasing for sufficiently large $x$. Since you were asked to prove that $f(x)$ increases at all positive values of $x$, it's not enough.

Here's a hint: what's the derivative of $g(x) = \frac{x}{1+ k x}$, where $k$ is a constant? Can you find a simplified expression? What can you say about its sign?

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