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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 ?

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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.

  • For higher dimensions and more complicated domain/constraints, do check out concept such Karush-Kuhn-Tucker conditions, convexity, coercive function. For non-smooth function, you might like to consider subgradient, branch and bound/ branch and price/ branch and cut. Optimization is a very broad topic and there is no silver bullet.
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    $\begingroup$ The 2 won't be there in differentiation. $\endgroup$ – Vyom Yadav Apr 4 at 4:42
  • $\begingroup$ thanks for pointing out the mistake $\endgroup$ – Siong Thye Goh Apr 4 at 4:43
  • $\begingroup$ You should also include the double (or more depending on situation) derivative test for maxima or minima with 0, it is obvious here but still, as a part of theory. $\endgroup$ – Vyom Yadav Apr 4 at 4:46

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