Given that :
The function is f(x) = 3x3 - 9x + 1.
By using,
Suppose x = c is the critical point of f(x) then,
If f ' (x ) > 0 to the left of x = c and f ' (x) < 0 to the right of x = c , then x = c is a relative maximum .
If f ' (x ) < 0 to the left of x = c and f ' (x) > 0 to the right of x = c , then x = c is a relative minimum .
An inflection point is a point on the graph at which the second derivative changes sign.
If f '' ( x ) > 0 then f(x) concave upwards.
If f'' (x) < 0 then f(x) concave downwards.
To find the critical points :
Differentiate the given equation with respect to x,
f' (x) = 9 x2 - 9
Set f' (x) = 0
9 x2 - 9 = 0
9 ( x2 - 1) = 0
x2 - 1 = 0
x2 = 1
Solve for x:
x = - 1, x = 1.
The critical points are x = - 1 and x = 1.
The domain of the given function is .
Combine the critical points x = - 1, x = 1 with the domain.
The functions monotone intervals are .
To check the sign of f' (x) = 9 x2 - 9 at each monotone interval :
Plug the extreme points x = - 1 in the given function.
f(-1) = 7 , to get y = 7.
Relative maximum = ( -1, 7)
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