Question

find a) any critical values and b) any relative extrema
f(x)=x² - 4x + 8
a) select the correct choice below and, if necessary, fill in the answer box within your choice
oa. the critical value(s) of the function is/are
(use a comma to separate answers as needed )
ob. the function has no critical values
b) select the correct choice below and, if necessary, fill in the answer box(es) within your choice
oa. the relative minimum point(s) is/are and the relative maximum point(s) is/are
(simplify your answers. type ordered pairs, using integers or fractions use a comma to separate answers as needed )
ob. the relative maximum point(s) is/are and there are no relative minimum points
(simplify your answer type an ordered pair, using integers or fractions. use a comma to separate answers as needed )
oc. the relative minimum point(s) is/are and there are no relative maximum points
(simplify your answer type an ordered pair, using integers or fractions. use a comma to separate answers as needed )
od. there are no relative minimum points and there are no relative maximum points

Explanation:

Step1: Find the derivative

The derivative of \(f(x)=x^{2}-4x + 8\) using the power rule \((x^n)^\prime=nx^{n - 1}\) is \(f^\prime(x)=2x-4\).

Step2: Find critical values

Set \(f^\prime(x) = 0\). So, \(2x-4=0\). Solving for \(x\):
Add \(4\) to both sides: \(2x=4\). Divide by \(2\): \(x = 2\).

Step3: Use the second - derivative test

Find the second - derivative \(f^{\prime\prime}(x)\). Differentiating \(f^\prime(x)=2x - 4\) gives \(f^{\prime\prime}(x)=2\).
Since \(f^{\prime\prime}(2)=2>0\), the function has a relative minimum at \(x = 2\).
Substitute \(x = 2\) into \(f(x)\): \(f(2)=2^{2}-4\times2 + 8=4-8 + 8=4\).

Answer:

a) A. The critical value(s) of the function is/are \(2\)
b) C. The relative minimum point(s) is/are \((2,4)\) and there are no relative maximum points