Increasing and Decreasing Functions

On an interval $I$:

• $f'(x) > 0$ for all $x \in I \Rightarrow f$ is strictly increasing on $I$.
• $f'(x) < 0$ for all $x \in I \Rightarrow f$ is strictly decreasing on $I$.

Method (the sign chart): solve $f'(x) = 0$ (and note where $f'$ is undefined); these critical points split the number line into intervals. Test the sign of $f'$ in each interval — a factored form like $(x-a)(x-b)(x-c)$ flips sign at each simple root. Where $f'$ is $+$, $f$ increases; where $-$, it decreases.

For $f$ a polynomial:

• Compute $f'(x)$ and factor it fully into linear (and irreducible-quadratic) factors.
• The simple real roots of $f'$ are the sign-change points. A product like $(x-a)(x-b)(x-c)$ is $+$ to the right of the largest root and alternates as you cross each root going left.
• A squared factor (double root) does NOT change sign — it touches zero and keeps the same sign on both sides.

Read off the increasing ($f'>0$) and decreasing ($f'<0$) intervals directly from the chart.

For a cubic $f$, $f'(x) = Ax^2 + Bx + C$ (with $A > 0$). Then:

• Discriminant $B^2 - 4AC < 0$ $\Rightarrow f'$ has no real roots $\Rightarrow f'(x) > 0$ for all $x$ $\Rightarrow f$ is strictly increasing on $\mathbb{R}$ (no turning points).
• Symmetrically, $A<0$ with $B^2-4AC<0$ gives $f'<0$ everywhere (strictly decreasing).

This is the standard way to prove 'increasing throughout the real line' or to impose 'no local extremum' as a parameter condition.

For $f(x) = \dfrac{a\sin x + b\cos x}{c\sin x + d\cos x}$, the quotient rule gives

• $f$ increasing for all $x$ $\iff ad - bc > 0$.
• $f$ decreasing for all $x$ $\iff ad - bc < 0$.

The same collapse happens for a simple rational $\dfrac{ax+b}{cx+d}$: $f'=\dfrac{ad-bc}{(cx+d)^2}$. A parameter version (e.g. $\dfrac{k\sin x+2\cos x}{\sin x+\cos x}$) turns 'strictly increasing' into a linear inequality in the parameter.

Differentiate with the product/chain rule, then isolate the factor whose sign is fixed:

• $e^{g(x)} > 0$ always, so in $f'(x) = e^{g(x)} \cdot (\text{stuff})$ the sign is the sign of stuff.
• $\dfrac{d}{dx}\log(u) = \dfrac{u'}{u}$; on the domain $u>0$, the sign is the sign of $u'$.
• For a composite $(f\circ g)'(x) = f'(g(x))\,g'(x)$, each factor's sign multiplies. Reduce to the product of the non-trivial factors and build their combined sign chart.

Standard collapses that make the derivative a single sinusoid:

• Triple angle: $3\sin x - 4\sin^3 x = \sin 3x$, so $f' = 3\cos 3x$.
• Power reduction: $\sin^4 x + \cos^4 x = 1 - \tfrac12\sin^2 2x$, giving $f' = -\sin 4x$.

Then read monotonicity from the sinusoid: $f' = 3\cos 3x > 0 \iff \cos 3x > 0$; $f' = -\sin 4x > 0 \iff \sin 4x < 0$. The longest increasing interval of $\sin(kx)$-type functions is the length of one rising quarter/half of the sinusoid — e.g. $f = \sin 3x$ rises on $\left(-\tfrac{\pi}{6},\tfrac{\pi}{6}\right)$, a run of length $\tfrac{\pi}{3}$.