Daily Math · 2026-07-06

Question 1

The function $f(x) = x^2 - 4x + 5$ has its minimum at:

Accepted Answer

$f'(x) = 2x - 4 = 0 \Rightarrow x = 2$. Since $f''(x) = 2 > 0$, this is a minimum.

Answer

$x = -2$

Answer

$x = 2$

Answer

$x = 4$

Answer

$x = 5$

Question 2

Calculați produsul $\log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{255} 256$.

Accepted Answer

Folosind schimbarea de bază, $\log_k (k+1) = \dfrac{\ln(k+1)}{\ln k}$. Produsul telescopează: $\dfrac{\ln 3}{\ln 2} \cdot \dfrac{\ln 4}{\ln 3} \cdots \dfrac{\ln 256}{\ln 255} = \dfrac{\ln 256}{\ln 2} = \log_2 256 = 8$.

Answer

$8$

Answer

$16$

Answer

$128$

Answer

$256$

Question 3

The last two digits of $7^{2024}$ are:

Accepted Answer

$7^2 = 49$, $7^4 = 49^2 = 2401 \equiv 1 \pmod{100}$. Since $2024 = 4 \cdot 506$, we have $7^{2024} = (7^4)^{506} \equiv 1^{506} = 1 \pmod{100}$. Last two digits: $01$.

Answer

$01$

Answer

$49$

Answer

$07$

Answer

$43$

Problems & worked solutions

Problem 1 — Applications of Derivatives

Problem 2 — Logarithms

Problem 3 — Number theory