Răspuns:
[tex]\boldsymbol {\red{x \in [0;2]}}[/tex]
Explicație pas cu pas:
[tex]35^x - 25^x \leq 28^x - 20^x + 21^x - 15^x[/tex]
[tex]5^x \cdot (7^x - 5^x) \leq 4^x \cdot (7^x - 5^x) + 3^x \cdot (7^x - 5^x)\\[/tex]
[tex]4^x \cdot (7^x - 5^x) + 3^x \cdot (7^x - 5^x) - 5^x \cdot (7^x - 5^x) \geq 0\\[/tex]
[tex](7^x - 5^x) \cdot (4^x + 3^x - 5^x) \geq 0 \ \ \big|:5^x\\[/tex]
[tex]\bigg[\bigg(\dfrac{7}{5}\bigg)^x - 1\bigg] \cdot \bigg[\bigg(\dfrac{4}{5}\bigg)^x + \bigg(\dfrac{3}{5}\bigg)^x - 1\bigg] \geq 0[/tex]
Astfel avem:
[tex]\bigg(\dfrac{7}{5}\bigg)^x \to strict \ cresc\breve{a}toare \Rightarrow f(x)=\bigg[\bigg(\dfrac{7}{5}\bigg)^x - 1\bigg] \to cresc\breve{a}toare[/tex]
Egalitatea are loc pentru x = 0
[tex]\bigg(\dfrac{4}{5}\bigg)^x \to strict \ descresc\breve{a}toare[/tex]
[tex]\bigg(\dfrac{3}{5}\bigg)^x \to strict \ descresc\breve{a}toare[/tex]
[tex]\Rightarrow g(x) =\bigg[\bigg(\dfrac{4}{5}\bigg)^x + \bigg(\dfrac{3}{5}\bigg)^x - 1\bigg] \to descresc\breve{a}toare[/tex]
Egalitatea are loc pentru x = 2 (deoarece 4² + 3² = 5²)
Tabelul de semn:
-∞....................0.........2..............+∞
[tex]f(x)----0+++++++[/tex]
[tex]g(x)-------0++++[/tex]
[tex]f(x)\cdot g(x) ++0--0++++[/tex]
De unde obținem:
[tex]\boldsymbol {x \in [0;2] }[/tex]
