The correct option is 4. (0,4)
Given inequalities are [tex]y\geq \frac{1}{2} x^{2} +4[/tex] and [tex]y< \sqrt{x+2}+6[/tex] .
Taking first inequality,
[tex]y\geq \frac{1}{2} x^{2} +4[/tex]
Or, [tex]y=\frac{1}{2} x^{2} +4[/tex]
now checking which option satisfy the above equation.
Taking option 1.(–2, 6)
on putting x= - 2 and y=6, we get
[tex]6= \frac{1}{2} (-2)^2+4[/tex]
[tex]6=6[/tex]
Hence option 1 (–2, 6) satisfy the equation.
Similarly checking for option 2 and 3 we find that these options totally not follow inequality, so option 3 and option 2 must not be the solution of given inequality.
Now option 4 (0, 4).
Putting x=0 and y = 4, we get
[tex]4=0+4\\4=4[/tex]
Hence option 4 also satisfy the first inequality.
Now we have two option to choose our final answer option 1 and option 4.
since option 4 lies in the shaded region so option 4 will be correct option.
Hence the correct option is 4. (0,4)
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