Question

Ma poate ajuta cineva cu aceste două exercitii la matematică vă rog ??

Multumesc din suflet !!

Answer 1

Explicație pas cu pas:

3.

$log_{[x]}(x+2)$

Avem condițiile de existență:

$(1) \ x+2 > 0\\x > -2 \iff x\in(-2,\infty)$

$(2) \ [x] > 0 \ \land \ [x]\neq 1$

$[x] > 0 \implies x \in [1, \infty)$

$[x]\neq 1\implies x \notin [1,2)$

$(1),(2) \implies x\in(-2,\infty) \cap[1,\infty) \setminus[1,2)=[2,\infty)\\\boxed{D=[2, \infty)}$

4.

$mx^2+2(m+1)x+9m+4 > 0 \iff m > 0 \ \land \ \Delta < 0$

$(1)\ m \in (0, \infty)$

$\Delta=4(m+1)^2-4m(9m+4)=4(m^2+2m+1)-4m(9m+4)=\\=4m^2+8m+4-36m^2-16m=-32m^2-8m+4\\\\-32m^2-8m+4 < 0$

$-32m^2-8m+4=0\\\Delta_m=64-4\cdot(-32)\cdot4=576\\m_{1,2}=\frac{8\pm24}{-64} \\m_1=\frac{-1}{2} \\m_2=\frac{1}{4}$

$\implies (2)\ m \in (-\infty, \frac{-1}{2} ) \cup(\frac{1}{4} ,\infty)$

$(1), (2) \implies \boxed{m \in (\frac{1}{4}, \infty)}$