# Find the difference between the maximum and the minimum value of the function

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 ?

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.
• The 2 won't be there in differentiation. – Vyom Yadav Apr 4 at 4:42
• thanks for pointing out the mistake – Siong Thye Goh Apr 4 at 4:43
• 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. – Vyom Yadav Apr 4 at 4:46