Step 1

Given

\(g(x)=-x^{2}+2x+6\)

To find out absolute max or min , let find out derivative g'(x)

\(g'(x)=−2x+2\)

Now set the derivative \(=0\) and solve for x

\(-2x+2=0\)

\(-2x=-2\)

\(x=1\)

Step 2

To check whether \(x=1\) is maximum or minimum we use second derivative test

\(g'(x)=−2x+2g''(x)=−2\)

second derivative is negative

so f(x) is maximum at \(x=1\)

There is no absolute minimum

Now find out absolute maximum value at \(x=1\)

Substitute \(x=1\) and find out g(1)

\(g(1)=-1^{2}+2(1)+6\)

\(g(1)=7\)

So absolute maximum value is 7

Absolute minimum does not exists