Question

# Find the absolute maximum value and the absolute minimum value, if any, of the function. g(x)=-x^{2}+2x+6

Functions
Find the absolute maximum value and the absolute minimum value, if any, of the function.
$$g(x)=-x^{2}+2x+6$$

2021-06-06

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