Find the difference between the maximum and the minimum value of the function $y = \sqrt{100-x^2}$ on the segment $[-4\sqrt2 ; 5\sqrt3]$.
Maximum value is $10$, minimum value is $5$. But what if $y$ is more complex ? Is there a different way to solve this problem, maybe using differentiation ?
Best Answer
For this question, the maximum value is clearly $10$ and the minimal value occur when the magnitude of $x$ is the largest.
Calculus approach:
$$y' = \frac{-x}{\sqrt{100-x^2}}$$
The stationary point is $0$ of which the correponding objective value is $10$.
After which, we can compare the objective value at the stationary point and the boundary point and conclude that the minimal value is $5$ and compute the difference.
You can also check that square root is an increasing function, and focus on finding the extrema of $100-x^2$ over the domain directly.