Maxima and Minima are one of the most common concepts in Differential Calculus. These two Latin words basically means maximum and minimum value of a function respectively, which is quite evident. Here, we assume our function to be continuous for its entire domain. Before knowing how to find maxima and minima, we should first learn about derivatives. Assuming that you all know how to find derivatives, let us go ahead and learn about some curves. What are curves?
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A curve is defined as onedimensional continuum. In figure 1, that curve is graph of a function \( f ~in~ x \)
Interval of a function plays a very important role to find extreme values of a function. If the interval for which the function \( f \)
domain.
Understanding local maxima and minima:
We may not be able to tell whether \( f(b) \)
 If \( f(a) \leq f(x) \)
for all \( x \) in \( P’s\) neighborhood (within the distance nearby \( P \) , where \( x = a \) ), \( f \) is said to have a local minimum at \( x = a \) .  If \( f(a) \geq f(x) \)
for all in \( P’s \) neighborhood (within the distance nearby \( P \) , where \( x = a \) ), \( f \) is said to have a local maximum at \( x = a \) .
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In the above example, \( B ~and~ D \)
Let us now take a point \( P \)
 If \( f'(a) = 0 \)
, the tangent drawn is parallel to \( x axis \) , i.e. slope is zero. There are three possible cases:
 The value of \( f \)
, when compared to the value of \( f \) at \( P \) , increases if you move towards right or left of \( P \) (Local minima: look like valleys)  The value of \( f \)
, when compared to the value of \( f \) at \( P \) , decreases if you move towards right or left of \( P \) (Local maxima: look like hills)  The value of \( f \)
, when compared to the value of \( f \) at \( P \) , increases and decreases as you move towards left and right respectively of \( P \) (Neither: looks like a flat land)
 If , the tangent drawn at has negative slope. The value of , when compared to the value of at , increases if you move towards left of and decreases if you move towards right of . So, in this case also, we can’t find any local extrema.
 If , the tangent drawn at has positive slope. The value of , when compared to the value of at , increases if you move towards right of and decreases if you move towards left of . So, in this case, we can’t find any local extrema.
 \( f’ \)
doesn’t exist at point \( P \) , i.e. the function is not differentiable at \( P \) . This normally happens when the graph of \( f \) has a sharp corner somewhere. All the three cases discussed in the previous point also hold true for this point.
To remember this, you can refer the Table 1.
Table 1: Various possibilities of derivatives of a function
Nature of f'(a)  Nature of Slope  Example  Local Extremum 
f'(a) > 0  Positive  Neither  
f'(a) < 0  Negative  Neither  
f'(a) = 0  Zero  Local Minimum
&; &; &; Local Maximum &; Neither &; &; 

Not Defined  Not Defined  Local Minimum
&; &; Local Maximum &; &; Neither 
In mathematics, a Critical point of differential function of real or complex variable is any value in its domain where its derivative is 0. We can hence infer from here that every local extremum is a critical point but every critical point need not be a local extremum. So, if we have a function which is continuous, it must have maxima and minima or local extrema. This means that every such function will have critical points. In case the given function is monotonic, the maximum and minimum values lie at the end points of the domain of the definition of that particular function.
Maxima and minima are hence very important concepts using which you can easily find the extreme values of a function. You can use these two values and where they occur for a function using the first derivative method or the second derivative method. To know and learn new concepts every day, download Byju’sthe learning app.
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