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Linearization, Mean-Value

lnxbiuregm
timer Asked: Dec 2nd, 2018

Question Description

This is calculus, Please answer the questions in the attached file. Work will need to be shown. The math is fairly simple.

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Let g be a function that is defined for all x, x ≠ 2, such that g(3) = 4 and the derivative of g is g′(x) = with x ≠ 2. 1. Find all values of x where the graph of g has a critical value. 2. For each critical value, state whether the graph of g has a local maximum, local minimum, or neither. You must justify your answers with a complete sentence. 3. On what intervals is the graph of g concave down? Justify your answer. 4. Write an equation for the tangent line to the graph of g at the point where x = 3. 5. Does this tangent line lie above or below the graph at this point? Justify your answer. Below are the answers. Please show work as to how to get the answers or if the answers below are incorrect fix them and show work. DBA 4.08 1. These points are where g'(x)=0, i.e. x=-4, x=4. 2. Near x=-4, g'(x) changes sign from – to +, so g(x) goes from decreasing to increasing, i.e. has a local minimum. The same is for x=4. 3. 𝑔′′ (𝑥) = 2𝑥(𝑥−2)−(𝑥 2 −16) (𝑥−2)2 = 𝑥 2 −4𝑥+16 , (𝑥−2)2 which is always positive (𝑥 ≠ 2). So 𝑔 is concave up on (−∞, 2) and on (2, +∞). 4. It is 𝑦 = 𝑔′ (3)(𝑥 − 3) + 𝑔(3) = −7(𝑥 − 3) + 4 = −𝟕𝒙 + 𝟐𝟓. 5. It is below the graph because we know the graph is concave up.
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