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INTRODUCTION
In mathematics, the maxima and minima of a function is called extrema. The highest point within the given range on the curve is called maxima point.The lowest point within the given range on the curves is called minima point. The extremum is found using the calculus of variation if an extremum consists of its own function. Maxima and minima can also be seen in sets but for now we will study it only for calculus.
CRITICAL POINT
The critical points for a function f(x) are those points of the domain where either f’(x) = 0 or doesn’t exist. At a critical point the value of function is a critical value.
For eg: f(x) = x2+8x+1
f’(x)=2x+8
F’(x)=0
2x+8 = 0
x = -4 are critical point of a function
CRITICAL NUMBERS OR STATIONARY POINT
The critical numbers of a function are those at which its first derivative is equal to 0.
Or we can call critical number stationary points which also says the same that the stationary points of a function are those points where the derivative of the function) is zero.
For eg: x3 − 6x2 + 9x
F’(x3 − 6x2 + 9x) =0
3x2-12x+9=0
3(x − 3)(x − 1)=0
X=3,1
Critical number or stationary point(3, −2) and (1, 2).
MAXIMA POINTS AND MINIMA POINTS
The highest point within the given range on the curve is called maxima point.
The lowest point within the given range on the curves is called minima point.
The local maximum: function changes from positive to zero to negative. The function is known as decreasing.
The local minimum function changes from negative, to zero, to positive. The function is increasing.
Let’s take one example
Q1: find relative min and max for f(x) = x3-3x2-9x+1?
Ans:f(x) = 3x2 -6x -9
= 3(x2-2x-3) = 0
x2-3x+x-3 = 0
(x+1)(x-3) = 0
x=-1,3
SECOND ORDER DERIVATIVE TEST FOR LOCAL EXTREMUM
- If n is odd , f has neither a local max. nor local min at x=xo.
- If n is even , f has local max if f(xo)<0 , f has local min if f(xo)>0.
Let’s take example of this
Ques1:- find local max and local min?
Ans:-f(x)=2x3-15x2+36x+18
F’(x)=6x2-30x+36 = 6(x2-5x+6) = 0
x2-5x+6 = 0
x(x-2)-3(x-2)= 0
(x-3)(x-2)= 0
x= 3,2
f”(x) = 12x-30
f”(2) = 24-30 = -6<0
f”(3) = 36-30 = 6>0
local max at x=2
local min at x=3
CONCLUSION
In this article we study the critical point , critical number which is also known as stationary point with examples. We will also talk about the first order derivatives test for relative local extrema and second derivatives test for relative local extrema.
In mathematics, the maxima and minima of a function is called extrema. The highest point within the given range on the curve is called maxima point.The lowest point within the given range on the curves is called minima point.
The extremum is found using the calculus of variation if an extremum consists of its own function. Maxima and minima can also be seen in sets but for now we will study it only for calculus.
Further we will do some questions related to maxima and minima.
