Daily math problems for August 24, 2026 — Descriptive Statistics & Sampling, Calculus, Algebra & more

The value appearing most often is $5$ (three times).

$f^{\mathrm{\prime }}(x)=-2x+4=0\Rightarrow x=2$. $f^{\mathrm{\prime }\mathrm{\prime }}(x)=-2<0$, confirming a local maximum.

Concatenation is associative; the empty string $\epsilon$ is the identity. No inverses exist (you can't "uncreate" letters), so this is a monoid (not a group). It's the free monoid on $\mathrm{\Sigma }$.

$S_3$ (or equivalently $D_3$) has $6$ elements and is non-abelian. All groups of order $1,2,3,4,5$ are abelian.

$2x=k\pi \Rightarrow x=k\pi \mathrm{/}2$. On $[0,2\pi )$: $0,\pi \mathrm{/}2,\pi ,3\pi \mathrm{/}2$. Four solutions.

$x^2=3\cdot 27=81\Rightarrow x=9$ (positive ratio).

Matrix multiplication is associative. The identity matrix $I_n$ is the identity. But the zero matrix has no inverse, so not all elements are invertible — it's a monoid but not a group.

$\bullet$ Simplify the exponent on the right: $i^{20}=(i^4{)}^5=1$, so $i^{20}+1=2$:

$$C_x^5=C_x^2$$

$\bullet$ Recall the combinatorial symmetry:

$$C_x^k=C_x^{x-k}$$

$\bullet$ Since $5\mathrm{\ne }2$, the symmetry forces $5+2=x$:

$$x=7$$

${\int }_0^{\pi \mathrm{/}2}{\sin }^{2}xdx={\int }_0^{\pi \mathrm{/}2}{\displaystyle \frac{1-\cos 2x}{2}}dx={\displaystyle \frac{1}{2}}{[x-\frac{\sin 2x}{2}]}_0^{\pi \mathrm{/}2}={\displaystyle \frac{\pi }{4}}$.

$f^{\mathrm{\prime }}(x)<0\iff 0<x<2$. So $f$ is decreasing on $(0,2)$. (Increasing on $(-\mathrm{\infty },0)$ and $(2,+\mathrm{\infty })$.)