Imagine you're on a roller coaster of numbers, and your goal is to find the highest and lowest points of the ride – the thrilling highs and comforting lows. Welcome to the world of Maxima and Minima! Just like figuring out the tallest hill or the deepest valley on your roller coaster adventure, in math, we're going to discover the peaks (Maxima) and valleys (Minima) of equations.
Where is a function at a high or low point? Calculus can help!
A maximum is a high point and a minimum is a low point:
Maxima and Minima
In a smoothly changing function a maximum or minimum is always where the function flattens out (except for a saddle point). Where does it flatten out? Where the slope is zero. The Derivative tells us!
To find the maxima or minima of a function using differentiation, you can follow these rules and steps:
1. Let's understand this with the help of an example:
f(x)=2x^{2}−8x+5
Identify the quadratic function: f(x)=2x^{2}−8x+5
Take the derivative of the function: f'(x)=4x−8
Set the derivative equal to zero and solve for x: 4x−8=0; 4x=8 ; x=2
The critical point is x=2.
So, by using differentiation, we found that the quadratic function f(x)=2x^{2}−8x+5 has a local minimum at x=2 with a corresponding ycoordinate of 9.
2. Now Let's take another example to understand something more interesting:
y = x^{3} − 6x^{2} + 12x − 5
The derivative is:
y' = 3x^{2} − 12x + 12
To find the critical points: y'=0
3x^{2} − 12x + 12=0
x= 2
Is it maxima or minima?
y''=6x12
y''(2)=6(2)12=0
If y''(x)=0, then this test fails.
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335 videos845 docs218 tests

Test: Basic Trigonometric Formulas Test  10 ques 
Test: Basic Trigonometric Formula Test  15 ques 
Test: Integration Basics Test  7 ques 

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