# Graph of Non-Monotonic Function

# Increasing function on interval

*f*(

*x*) is increasing in interval (

*a*,

*b*) if for all

*x*

_{1},

*x*

_{2}âˆˆ (

*a*,

*b*), if

*x*

_{1}<

*x*

_{2}â‡’

*f*(

*x*

_{1}) â‰¤

*f*(

*x*

_{2}).

In the given graph of function if we draw any tangent line, it makes either acute angle with positive direction of the

*x*-axis or it is parallel to the*x*-axis, i.e. slope of tangent at any point is non-negative. Also slope of tangent is equal to derivative of function at that point, i.e.,*f*â€²(*x*) â‰¥ 0 for all*x*in the interval (*a*,*b*). So when*f*â€²(*x*) â‰¥ 0 for all*x*in the interval (*a*,*b*) of consideration,*f*(*x*) is said to be increasing.For example, for

*f*(*x*) =*x*^{3},*f*â€²(*x*) = 3*x*^{2}â‰¥ 0, i.e.*f*(*x*) is an increasing function on*R*(domain of function).# Strictly increasing function on interval

*f*(

*x*) is strictly increasing in interval (

*a*,

*b*) if for all

*x*

_{1},

*x*

_{2}âˆˆ (

*a*,

*b*), if

*x*

_{1}<

*x*

_{2}â‡’

*f*(

*x*

_{1}) <

*f*(

*x*

_{2}).

In the given graph of function if we draw any tangent line, it makes acute angle with positive direction of

*x*-axis, i.e., slope of tangent at any point is positive, i.e.,*f*â€²(*x*) > 0 for all*x*in the interval (*a*,*b*). So when*f*â€²(*x*) > 0 for all*x*in the interval (*a*,*b*) of consideration,*f*(*x*) is said to be strictly increasing.For example, for

*f*(*x*) = log*x*,*f*â€²(*x*) = 1/*x*> 0 (for*x*> 0), i.e.,*f*(*x*) is strictly increasing function on (0, âˆž) (domain of function).# Decreasing function on interval

*f*(

*x*) is decreasing in interval (

*a*,

*b*) if for all

*x*

_{1},

*x*

_{2}âˆˆ (

*a*,

*b*), if

*x*

_{1}<

*x*

_{2}â‡’

*f*(

*x*

_{1}) â‰¥

*f*(

*x*

_{2}).

In the given graph of function if we draw any tangent line, it makes either obtuse angle with positive direction of the

*x*-axis or it is parallel to the*x*-axis, i.e., slope of tangent at any point is non-positive, i.e.,*f*â€²(*x*) â‰¤ 0 for all*x*in the interval (*a*,*b*). So when*f*â€²(*x*) â‰¤ 0 for all*x*in the interval (*a*,*b*) of consideration,*f*(*x*) is said to be increasing.For example, for

*f*(*x*) = sin*x*â€“*x*,*f*â€²(*x*) = cos*x*â€“ 1 â‰¤ 0 (as â€“ 1 â‰¤ cos*x*â‰¤ 1), i.e.*f*(*x*) is a decreasing function on*R*(domain of function).# Strictly decreasing function on interval

*f*(

*x*) is strictly decreasing in interval (

*a*,

*b*) if for all

*x*

_{1},

*x*

_{2}âˆˆ (

*a*,

*b*), if

*x*

_{1}<

*x*

_{2}â‡’

*f*(

*x*

_{1}) >

*f*(

*x*

_{2})

In the given graph of function if we draw any tangent line, it makes obtuse angle with positive direction of the

*x*-axis, i.e., slope of tangent at any point is positive, i.e.,*f*â€²(*x*) < 0 for all*x*in the interval (*a*,*b*). So when*f*â€²(*x*) < 0 for all*x*in the interval (*a*,*b*) of consideration,*f*(*x*) is said to be strictly decreasing.For example, for

*f*(*x*) =*e*^{â€“x},*f*â€²(*x*) = â€“*e*^{â€“x}< 0 (as*e*^{â€“x}> 0 for all real*x*), i.e.,*f*(*x*) is strictly decreasing function on*R*(domain of function).# Increasing/decreasing at a point

Let

*x*_{0}be a point in the domain of definition of a real- valued function*f*. Then*f*is said to be increasing, strictly increasing, decreasing or strictly decreasing at*x*_{0}if there exists an open interval*I*containing*x*_{0}such that*f*is increasing, strictly increasing, decreasing or strictly decreasing, respectively, in*I*.For example, a function

*f*is said to be increasing at*x*_{0}if there exists an interval*I*= (*x*_{0}â€“*h*,*x*_{0}+*h*),*h*> 0 such that for*x*_{1},*x*_{2}âˆˆ*I*and*x*_{1}<*x*_{2}in*I*â‡’*f*(*x*_{1}) â‰¤*f*(*x*_{2}).