Increasing and Decreasing Functions

Increasing and decreasing functions describe how the value of a function changes as the input value increases. This topic is an important part of Applications of Derivatives and helps students study graph behaviour, monotonicity, and later ideas such as maxima and minima.

A function f(x) is said to be an increasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

Strictly Increasing Function:

• If x₁ < x₂ ⇒ f(x₁) < f(x₂)

A function f(x) is said to be a decreasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

Strictly Decreasing Function:

• If x₁ < x₂ ⇒ f(x₁) > f(x₂)

A function f is said to be constant on I if f(x) = c for every x ∈ I, where c is a constant.

A function f is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

Let x₀ be a point in the domain of a real-valued function f.

The function f is said to be increasing at x₀ if there exists an open interval containing x₀ in which f is increasing.

Similarly, f is said to be decreasing at x₀ if there exists an open interval containing x₀ in which f is decreasing.

Let f be continuous on [a, b] and differentiable on (a, b).

Then:

1. Increasing Function

If \[ \boxed{f'(x) \geq 0} \] for every x ∈ (a, b), then f is increasing on [a, b].

If \[ f'(x) > 0 \] throughout the interval, then f is strictly increasing.

2. Decreasing Function

If \[ \boxed{f'(x) \leq 0} \] for every x ∈ (a, b), then f is decreasing on [a, b].

If \[ f'(x) < 0 \] throughout the interval, then f is strictly decreasing.

3. Constant Function

If \[ \boxed{f'(x) = 0} \] for every x ∈ (a, b), then f is constant on [a, b].

As we move from left to right on the graph:

• if the y-values increase, the function is increasing;
• if the y-values decrease, the function is decreasing;
• if the y-values remain constant, the function is constant.

Strictly increasing function (i)	Strictly decreasing function (ii)	Neither increasing nor decreasing function (iii)

For a differentiable function f(x):

Step 1: Find \[ f'(x). \]

Step 2: Solve \[ f'(x) = 0. \]

Also note any points in the domain where f'(x) is undefined.

Step 3:These points divide the domain into intervals.

Step 4: Determine the sign of f'(x) in each interval.

Step 5:

•  if \[f'(x) > 0,\] f is increasing;
• if \[f'(x) < 0,\] f is decreasing.

Step 6: Write the intervals clearly.

Show that the function \[f\] given by \[f(x) = x^3 - 3x^2 + 4x, x \in \mathbf{R}\] is increasing on R.

Solution: Note that

\[= 3(x^2 - 2x + 1) + 1\]

\[= 3(x - 1)^2 + 1 > 0\], in every interval of R

Therefore, the function \[f\] is increasing on R.

Find the intervals in which the function \[f\] given by \[f(x) = 4x^3 - 6x^2 - 72x + 30\] is (a) increasing (b) decreasing.

Solution: 

Step 1: Differentiate the function

or \[f'(x) = 12x^2 - 12x - 72\]

\[= 12(x^2 - x - 6)\]

\[= 12(x - 3)(x + 2)\]

Step 2: Find critical points

\[f'(x) = 0\] gives \[x = -2, 3\]. The points \[x = -2\] and \[x = 3\] 

These two x-values cut the real line into 3 intervals:

• (−∞,−2)

• (−2,3)

• (3,∞)

In the intervals \[(-\infty, -2)\] and \[(3, \infty), f'(x)\] is positive while in the interval \[(-2, 3)\], \[f'(x)\] is negative.

Step 3: Sign of f′(x) on each interval

\[f\] is neither increasing nor decreasing in R.

• Increasing means output does not decrease as input increases.
• Strictly increasing means output always increases.
• Decreasing means output does not increase as input increases.
• Monotonic means either increasing or decreasing on an interval.
• f′(x) > 0 implies increasing, f′(x) < 0 implies decreasing, and f′(x) = 0 on an interval implies constant behaviour.
• If \[ f'(x) = 0 \] throughout an interval, the function is constant on that interval.
• A single point where \[ f'(x) = 0 \] does not necessarily make the function constant.
• To find intervals of increase or decrease, find the zeros of f'(x), divide the domain into intervals, and check the sign of f'(x).
• A function that is increasing or decreasing on an interval is called monotonic on that interval.
• A function may be increasing on one interval and decreasing on another; in that case it is not monotonic on its entire domain.