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The first derivative helps you to identify all the critical points on the curve because when dy/dx = 0, the there will a turn point at that location ie.. going up -> turn -> going down.
In the curve, there might have alot of critical points eg. y = x^6 + x^5 + x^4 +etc. However, there will be only one extreme maxima and/or one extreme minima.
thanx,
no i understand this point, what i`m asking for is "how does finding the derivative do that? how does finding the derivative find the critical points?"
Taking derivative gives u a function that defines the slope of a curve at every point (generally speaking)... extremas exist at points which have 0 slope.
as fsahmed said, the derivative is a function of the slope of the tangent at each point on the curve, critical points are the points on the curve whose tangent's slope is 0, that is points whose tangent is parallel to the x-axis... so what you do is finding the points at which the derivative=0.
each extreme value is a critical point but not each critical point is necesseraly an extreme value... is that clear?
Adding wat asoom has said that:
critical point is the specified point where curve changes its slope.
in much simpler word, I should say
the point where linearity is diturbed.
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