Lab 6 – Extreme Values



2. There are two SageMath apps below. The first app plots the graph of \(f(x)\) and its derivative in the interval \( [x_{min}, x_{max}] \). It also evaluates the function \( f \) at \( x \). The second app is used to find the roots of \( f'(x) \). Note that you will need to compute the derivative of \( f(x) \) to use this app. When searching for a particular root, put a lower bound into “Left” and an upper bound into “Right”. In other words, make sure your desired root is the only one in the interval [Left, Right].

(a) Use the first SageMath app below to find \( f(a) \) and \( f(b) \) for your given function \( f(x) \) and given interval \( [a,b] \).
(b) Use the second SageMath app below to find the critial points of \( f(x) \) on \( [a,b] \). Note that you will need to compute \( f'(x) \) first before using this app.
(c) For each critical point \( c \) found in part (b), compute \( f(c) \) using the first SageMath app.
(d) Find the absolute maximum and absolute minimum of \( f(x) \) on the interval \( [a,b] \).

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