Physics and Mathematics
Decreasing Function
1. Concept Overview
A function describes how one quantity changes with respect to another.
A function can decrease, meaning its output becomes smaller as the input increases.
In calculus, the decreasing nature of a function is determined using the first derivative.
2. Increasing vs. Decreasing – Calculus Interpretation
Let a function be [f(x)] defined on an interval [I].
Decreasing Function
A function is decreasing on an interval if:
[x_1 < x_2 \implies f(x_1) > f(x_2)]
Calculus Condition
If the function is differentiable, then:
[f'(x) < 0] for all [x]Â in the interval
This guarantees that the slope of the tangent at every point is negative → the graph falls as x increases.
3. Strictly Decreasing vs. Non-Increasing
Strictly Decreasing
[f'(x) < 0 \ \forall x \in I]
The function is continuously sloping downward with no flat (horizontal) tangents.
Non-Increasing (Decreasing or Constant)
[f'(x) \le 0 \ \forall x \in I]
A function may include:
- downward slopes, and/or
- flat segments [f'(x)=0],
but never upward slopes.
Difference:
- Strictly decreasing → tangent slope always negative
- Non-increasing → slope is negative or zero
4. Tangent Interpretation
If the tangent at every point is sloping downwards such that:
[\theta] (angle with positive x-axis) [> 90^\circ][\quad] and [\quad][\theta < 180^\circ]
then the slope satisfies:
[
\boxed{m = \tan\theta < 0}
]
Hence the function is decreasing.
5. Conceptual Questions With Solutions
1. If a function has [f'(x) < 0] for all x, what does this tell you?
It means the function is strictly decreasing on that interval because the tangent slope is negative everywhere.
2. Can a decreasing function have [f'(x)=0] at some points?
Yes. The function still remains non-increasing. A zero derivative at isolated points does not make it increase.
3. If [f'(x) \le 0] everywhere, what type of function is it?
It is a non-increasing function (it may be decreasing or constant, but it never rises).
4. If [f'(a)] is negative at one point, must the function be decreasing on the entire interval?
No. Decreasing behaviour must be checked on the whole interval, not at a single point.
5. Can a function be decreasing even if it is not differentiable at some points?
Yes. If for any [x_1 < x_2], we have [f(x_1) > f(x_2)], the function is decreasing, even with corners or cusps.
6. If [f'(x) = -(x^2 + 1)], what can we conclude?
Since [x^2 + 1] is always positive, [f'(x)] is always negative. Thus, the function is strictly decreasing everywhere.
7. What does a negative tangent slope indicate geometrically?
It indicates the tangent makes an angle [\theta] with the positive x-axis such that [\theta > 90^\circ], giving [\tan\theta < 0]. The graph slopes downward.
8. Are all decreasing functions invertible?
Yes. A strictly decreasing (or strictly monotonic) function is one–one and therefore invertible.
9. Can a quadratic function be decreasing everywhere?
No. A quadratic changes slope; it can only be decreasing on one interval, not the entire real line.
10. If [f'(x)=0] on an entire interval, what does it imply?
The function is constant on that interval, which is a special case of non-increasing behaviour.
11. If [f'(x)] remains negative on an interval, what can we conclude?
The function is strictly decreasing throughout that interval.
12. When does the derivative test fail for decreasing behaviour?
When the function is not differentiable at some points. The monotonicity must then be checked from the definition.
13. Is a decreasing function always concave down?
No. Decreasing relates to the first derivative; concavity relates to the second derivative. They are independent.
14. If the slope of tangent is m = -4 at some x, what does it mean?
It means the function is decreasing at that point because slope is negative.
15. What does [f'(x) = \tan\theta] tell us about decreasing behaviour?
If the tangent angle [\theta] lies between [90^\circ] and [180^\circ], then [\tan\theta < 0], meaning the function is decreasing at that point.
6. FAQ / Common Misconceptions
1. Does decreasing mean the function’s values are negative?
No. Decreasing refers to the trend, not the sign of the function values.
2. Must a decreasing function have a negative derivative at all points?
No. It may have points where [f'(x)=0] but still be non-increasing.
3. If the function is decreasing at one point, is it decreasing everywhere?
No. Decreasing behaviour must be checked over the whole interval.
4. Can a decreasing function later become increasing?
Yes. Functions may have separate intervals of decrease and increase.
5. If [f'(x)=0], does the function have a turning point?
Not necessarily. It could be flat but still part of a non-increasing function.
6. Does decreasing imply [f”(x) < 0]?
No. Concavity and monotonicity are not the same.
7. Can a decreasing function have a local maximum?
No. A strictly decreasing function cannot have any local maximum inside the interval.
8. Is a decreasing function always differentiable?
No. Corners and nondifferentiable points may exist in decreasing functions.
9. Does negative slope always imply decreasing?
Yes. A negative slope indicates falling behaviour at that point.
10. Can a decreasing function become constant?
Yes. That makes it non-increasing, not strictly decreasing.
7. Example with Step by Step Solutions
Example 1 — Polynomial
Determine where [f(x)=x^{3}-3x] is decreasing.
Step-by-step solution:
- Compute derivative: [\dfrac{d}{dx}f(x)][=3x^{2}-3 ][= 3(x^{2}-1)].
- Find critical points: solve [3(x^{2}-1)=0] ⇒ [x^{2}=1] ⇒ [x=\pm 1].
- Make a sign chart for [3(x^{2}-1)] on intervals [(-\infty,-1),\ (-1,1),\ (1,\infty)]:
- Pick [x=-2]: [3(4-1)=9>0] → derivative positive.
- Pick [x=0]: [3(0-1)=-3<0] → derivative negative.
- Pick [x=2]: [3(4-1)=9>0] → derivative positive.
- Conclusion: [f(x)] is decreasing on (-1,1) (derivative < 0 there).
Example 2 — Logarithmic / Linear mix
Find where [f(x)=\ln x – \dfrac{x}{2}] (domain [x>0]) is decreasing.
Step-by-step solution:
- Compute derivative for [x>0]: [\dfrac{d}{dx}f(x)][=\dfrac{1}{x}-\dfrac{1}{2}].
- Solve [\dfrac{1}{x}-\dfrac{1}{2}<0] for decreasing:
[\dfrac{1}{x} < \dfrac{1}{2} \Rightarrow x > 2].
(Equality at [x=2] gives stationary point.) - Test: for [x=3], derivative = [1/3 – 1/2 = -1/6 < 0]. For [x=1], derivative = [1 – 1/2 = 1/2 > 0].
- Conclusion: [f(x)] is decreasing on (2,\infty) (and increasing on (0,2)).
Example 3 — Exponential with square
Determine decreasing intervals for [f(x)=e^{-x^{2}}].
Step-by-step solution:
- Compute derivative using chain rule: [\dfrac{d}{dx}f(x) ][= -2x e^{-x^{2}}].
- Solve [ -2x e^{-x^{2}} < 0 ]. Note [e^{-x^{2}}>0] always, so sign is that of [ -2x ]:
- [ -2x < 0 ] ⇔ [ x > 0 ].
- Therefore derivative negative for [x>0]; derivative positive for [x<0]. At [x=0] derivative = 0.
- Conclusion: [f(x)] is decreasing on [(0,\infty)] and increasing on [(-\infty,0)] (global maximum at [x=0]).
Example 4 — Rational function
Find where [f(x)=\dfrac{x}{1+x^{2}}] is decreasing.
Step-by-step solution:
- Differentiate (quotient rule):
[\dfrac{d}{dx}f(x)=\dfrac{(1+x^{2})\cdot 1 – x\cdot 2x}{(1+x^{2})^{2}} = \dfrac{1 – x^{2}}{(1+x^{2})^{2}}]. - Denominator is always positive, so sign depends on numerator [1-x^{2}]. For decreasing we need [1-x^{2}<0] ⇒ [x^{2}>1] ⇒ [|x|>1].
- Critical points at [x=±1], where derivative = 0.
- Conclusion: [f(x)] is decreasing on [(-\infty,-1)\cup(1,\infty)] and increasing on [(-1,1)].
Example 5 — Implicit curve (circle) — local decreasing of y as a function of x
Consider the circle [x^{2}+y^{2}=25]. For the upper semicircle (take [y>0]), determine for which x the function defined by [y=y(x)] is decreasing.
Step-by-step solution:
- Differentiate implicitly w.r.t [x]: [2x + 2y,\dfrac{dy}{dx} = 0] ⇒ [\dfrac{dy}{dx} = -\dfrac{x}{y}].
- On the upper semicircle we have [y = +\sqrt{25 – x^{2}}] > 0 for [-5<x<5]. So sign of [\dfrac{dy}{dx}] is opposite to sign of [x]:
- If [x>0], then [ -\dfrac{x}{y} < 0 ] (since y>0) → derivative negative.
- If [x<0], then derivative positive. At [x=0], derivative = 0.
- Conclusion: On the upper semicircle [(y>0)], the vertical coordinate [y(x)] is decreasing for x\in [(0,5)] and increasing for x\in [(-5,0)]. (Symmetric results hold on the lower semicircle with sign flips.)
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