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 = -2/3So 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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