Monotonicity flash cards

1. 1.If $f'(x) > 0$ for all $x$ in $(a,b)$, what can you conclude about $f(x)$ on $(a,b)$, and what is the necessary caveat regarding the domain?

   $f(x)$ is strictly increasing on $(a,b)$. Caveat: monotonicity must be checked on each interval of continuity/domain separately — a function can be increasing on two disjoint intervals but NOT increasing on their union if there's a jump or the point is excluded (e.g., $f(x)=-1/x$ is increasing on $(-\infty,0)$ and on $(0,\infty)$ separately, but not on $(-\infty,0)\cup(0,\infty)$ as a whole).

   Hint: Domain splitting matters

2. 2.State the sign test used to determine intervals of increase/decrease of $f(x)$ using $f'(x)$.

   Find $f'(x)$, locate critical points (where $f'(x)=0$ or undefined), mark them on the number line, then test the sign of $f'(x)$ in each sub-interval: $f'(x)>0 \Rightarrow$ increasing; $f'(x)<0 \Rightarrow$ decreasing.

   Hint: Wavy curve / sign scheme method

3. 3.How do you prove an inequality like $\sin x < x$ for $x>0$ using monotonicity?

   Define $f(x)=x-\sin x$. Then $f'(x)=1-\cos x \ge 0$ for all $x$, so $f$ is increasing on $[0,\infty)$. Since $f(0)=0$, for $x>0$ we get $f(x)>f(0)=0$, i.e. $x-\sin x>0 \Rightarrow \sin x < x$.

   Hint: Construct auxiliary function, use $f(0)$

4. 4.For a strictly increasing continuous function $f$, if $f(g(x)) < f(h(x))$, what can you conclude about $g(x)$ and $h(x)$? What changes if $f$ is strictly decreasing?

   If $f$ is strictly increasing: $f(g(x)) < f(h(x)) \Rightarrow g(x) < h(x)$ (inequality direction preserved). If $f$ is strictly decreasing: the inequality reverses, giving $g(x) > h(x)$.

   Hint: Increasing preserves, decreasing flips

5. 5.What is the condition on $f'(x)$ for $f(x)$ to be monotonic (non-decreasing) on an interval, as opposed to strictly increasing? Give an example distinguishing the two.

   For non-decreasing (monotonic increasing): $f'(x) \ge 0$ throughout the interval, with $f'(x)=0$ only at isolated points (not on any sub-interval). Example: $f(x)=x^3$ has $f'(x)=3x^2 \ge 0$ with equality only at $x=0$ (isolated point), so $f$ is strictly increasing on $\mathbb{R}$ even though $f'(0)=0$.

   Hint: Zero at isolated points still allows strict

6. 6.Find the intervals of increase and decrease of $f(x)=x^3e^{-x^2}$ on $\mathbb{R}$.

   $f'(x)=e^{-x^2}(3x^2-2x^4)=x^2e^{-x^2}(3-2x^2)$. Since $x^2e^{-x^2}\ge0$ always, the sign of $f'$ matches the sign of $3-2x^2$. So $f$ is increasing on $\left(-\sqrt{3/2},\sqrt{3/2}\right)$ and decreasing on $\left(-\infty,-\sqrt{3/2}\right)\cup\left(\sqrt{3/2},\infty\right)$.

   Hint: Factor out the always-nonnegative part of $f'$ before analyzing signs.

7. 7.Determine the intervals of monotonicity of $f(x)=x^2\ln x$ for $x>0$.

   $f'(x)=2x\ln x+x=x(2\ln x+1)$. Since $x>0$, the sign of $f'$ equals the sign of $2\ln x+1$, which vanishes at $x=e^{-1/2}$. So $f$ is decreasing on $(0,e^{-1/2})$ and increasing on $(e^{-1/2},\infty)$.

   Hint: Factor out $x$ first, then solve $2\ln x+1=0$.

8. 8.Find the intervals of increase and decrease of $f(x)=\dfrac{x}{\ln x}$ for $x>0,\,x\ne1$.

   $f'(x)=\dfrac{\ln x-1}{(\ln x)^2}$, and the denominator is always positive. So the sign of $f'$ equals the sign of $\ln x-1$, giving $f$ decreasing on $(0,1)\cup(1,e)$ and increasing on $(e,\infty)$.

   Hint: Use the quotient rule; the denominator's square is never a sign issue.

9. 9.Determine where $f(x)=x^x$ ($x>0$) is increasing and where it is decreasing.

   Write $f(x)=e^{x\ln x}$, so $f'(x)=x^x(\ln x+1)$. Since $x^x>0$, the sign of $f'$ matches $\ln x+1$, which vanishes at $x=1/e$. So $f$ is decreasing on $(0,1/e)$ and increasing on $(1/e,\infty)$, with global minimum $e^{-1/e}$.

   Hint: Take logarithms first to differentiate $x^x$ easily.

10. 10.Find the intervals of monotonicity of $f(x)=x^{1/x}$ for $x>0$.

    Let $y=\ln f=\dfrac{\ln x}{x}$, so $y'=\dfrac{1-\ln x}{x^2}$ and $f'=f\cdot y'$. Since $f>0$, the sign of $f'$ matches $1-\ln x$, positive for $xe$. Hence $f$ increases on $(0,e)$ and decreases on $(e,\infty)$, with maximum value $e^{1/e}$.

    Hint: Take logs to convert the power into a product before differentiating.

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