Question

sa se determine m apartine R astfel incat parabolele asociate functiilor f (x)=x^2-2x-4 si g (x)=mx^2-2mx-6 sa aibă acelasi varf.

Answer 1

$\hbox{Varful parabolei ecuatiei de gr. 2 are formula:} \\\\ V(-\frac{b}{2a}; - \frac{\Delta}{4a}}) \\\\\\ f(x)=x^2-2x-4 \\ . \ \downarrow \\\\ V(-\frac{-2}{2*1};-\frac{4-4*(-4)}{4*1}}) \ \ \textless \ =\ \textgreater \ \ V(1;- \frac{4+16}{4}) \to V(1;-5) \\\\\\\\ g(x)=mx^2-2mx-6 \\ . \ \downarrow \\\\ V(- \frac{-2m}{2*m};-\frac{4m^2-4*m*(-6)}{4m}) \ \ \textless \ =\ \textgreater \ \ V(1; -\frac{4m(m+6)}{4m}) \\\\\\\ V(1;-5)= V(1;-(m+6)) \\\\\\ \Longrightarrow m+6=5 \longrightarrow \boxed{m=-1}$

Answer 2

 

Cele două parabole au același vârf, deci :

[tex]\it f_{max} =g_{max} \Rightarrow -\dfrac{4+16}{4} =-\dfrac{4m^2+24}{4m} \Rightarrow 5 = \dfrac{4m(m+6)}{4m} \Rightarrow \\\;\\ \it5 = m+6 \Rightarrow 5-6 =m \Rightarrow m =-1. [/tex]