Question

Use the function f(x) = x^2 − 2x + 8 and the graph of g(x) to determine the difference between the maximum value of g(x) and the minimum value of f(x).*Question in image*Please explain ~

Answer 1

Notice that f(x) corresponds to a parabola that opens upwards on the plane; therefore, its only critical point is a minimum. Solve f'(x)=0 to find the minimum value of f(x), as shown below

$\begin{gathered} f^{\prime}(x)=0 \\ \Rightarrow2x-2=0 \\ \Rightarrow x=1 \\ \Rightarrow(1,f(1))=(1,7)\rightarrow\text{ minimum} \end{gathered}$

On the other hand, from the image, the maximum value of g(x) is at (3,12).

Therefore, the difference between the maximum value of g(x) and the minimum value of f(x) is

$12-7=5$

The answer is 5.