10_Ex 2.10

Exercise Questions

Questions	Question Links
Q1. Find two positive integers…	2.10 Q-1
Q2. Divide 20 into two parts so that sum…	2.10 Q-2
Q3. Find two positive integers whose sum…	2.10 Q-3
Q4. The perimeter of a triangle is 16 cm…	2.10 Q-4
Q5. Find the dimension of a rectangle…	2.10 Q-5
Q6. Find the lengths of the sides…	2.10 Q-6
Q7. A box with a square base and open top…	2.10 Q-7
Q8. Find the dimension of rectangular garden	2.10 Q-8
Q9. An open tank of square base of side $x$ …	2.10 Q-9
Q10. Find the dimension of the rectangle…	2.10 Q-10
Q11. Find the point on the curve $y=x^{2}-1$ …	2.10 Q-11
Q12. Find the point on the curve $y=x^{2}+1$ …	2.10 Q-12

Overview

This exercise involves solving optimization problems where we are asked to maximize or minimize certain quantities under specific constraints. These problems are useful in a wide range of applications, such as maximizing profit, minimizing cost, or finding the best dimensions for physical objects like rectangles, boxes, and triangles.

Key Concepts

Optimization Problems: Problems where we maximize or minimize a given function subject to constraints.
Differentiation: Used to find the critical points by setting the first derivative equal to zero.
Second Derivative Test: Helps determine whether a critical point corresponds to a maximum or minimum by analyzing the concavity of the function.

Important Formulas

First Derivative: Used to find the critical points of a function $f(x)$ :

f'(x) = 0

Second Derivative: Used to test the nature of the critical points:

f''(x) < 0 \quad \text{(local maximum)}, \quad f''(x) > 0 \quad \text{(local minimum)}

Tips and Tricks

For problems involving perimeter and area, the key is to express the constraint (such as a fixed perimeter) in terms of one variable and then differentiate the objective function.
In problems where you’re asked to maximize or minimize the product or sum, first establish the equation for the function and simplify it before differentiating.
Always check that the second derivative is negative for a maximum or positive for a minimum when applying the second derivative test.

Summary

In this exercise, we solved several optimization problems using derivatives. For each problem:

We set up a function to represent the quantity to be maximized or minimized.
We found the critical points by setting the first derivative equal to zero.
We used the second derivative to confirm whether the critical point represented a maximum or minimum.

Reference

By Sir Shahzad Sair