Dawson Downs

2022-07-20

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

juicilysv

Expert

Step 1
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 $\frac{22000x}{\left(11+{x}^{2}{\right)}^{2}}$.
Step 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.

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