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Critical Points

Critical points of a function are the values in its domain where its derivative is zero. Since the derivative of a function is the slope of its graph, hence critical points are the points where the function is neither increasing nor decreasing, but is constant at that point. Note that the function may be increasing/decreasing just before or just after the point, but it is not changing only exactly at that point.

Critical points of a function are also present where its derivative is not defined.

An interesting fact is that critical points are the points where the graph of the function has a relative maximum or a relative minimum. Absolute maximum's/minimum's are also present only at the critical points

Example

Let a function be f(x) = 3x^2 + 4x + 5. Its critical points are computed as follows:

Step 1: Compute the derivative of the function

f `(x) = d/dx [ 3x^2 + 4x + 5 ]
By sum rule,
f `(x) = d/dx (3x^2) + d/d (4x) + d/dx (5)
Factor out the constants
f `(x) = 3 d/dx (x^2) + 4 d/d (x) + d/dx (5)
Derivative of a constant (that is a number) is zero:
f `(x) = 3 d/dx (x^2) + 4 d/d (x) + 0
By power rule,
f `(x) = 3 (2x) + 4 (1) + 0
Simpilfy the expression:
f `(x) = 6x + 4

Step 2: Equate the derivative to zero and solve the equation for 'x'

6x + 4 = 0
x = -4/6 = -2/3
So the function f(x) = 3x^2 + 4x + 5 has a critical point at x = -2/3. Graphically, this means that there is a relative maximum or a relative minimum at this point. The following graph of f(x) has a relative minimum (vertex of the parabola) at x = -2/3

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