# How to find where a function is increasing at the greatest rate

Given the function $f(x) = \frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate?

Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.

The derivative of $f(x)$ is $\frac{22000x}{(11+x^2)^2}$

Applying the first derivative test, the critical number is $\sqrt{\frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $\sqrt{\frac{11}{3}}$ is the answer.

• What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively. – Zach Stone Dec 18 '15 at 4:30
• "...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular) – The Chaz 2.0 Dec 18 '15 at 5:06
• @The Chaz 2.0, Edited! – hmir Dec 18 '15 at 5:08
• Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative. – The Chaz 2.0 Dec 18 '15 at 5:11
• It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $\sqrt{\frac{11}{3}}$: $$f\left(\sqrt{\dfrac{11}{3}+0.1}\right)-f\left(\sqrt{\dfrac{11}{3}}\right)\approx5.07900\\f\left(\sqrt{\dfrac{11}{3}}\right)-f\left(\sqrt{\dfrac{11}{3}-0.1}\right)\approx5.14875$$ – Piquito Apr 5 '18 at 12:07

The derivative of $f(x)$ is $\frac{22000x}{(11+x^2)^2}$
Applying the first derivative test, the critical number is $\sqrt{\frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $\sqrt{\frac{11}{3}}$ is the answer.