Question

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

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asked 2021-06-05
Find the absolute maximum value and the absolute minimum value, if any, of the function.
\(g(x)=-x^{2}+2x+6\)

Answers (1)

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

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