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The maximum rate of change is then
which occurs in the direction of
Answer:
Step-by-step explanation:
Given a function f, whose derivatives are f' and f'', a value x is a critical point if f'(x) =0. A value x is a minimum of f if it is a critical point and f''(x) >0 and it is maximum if f''(x)<0. We will perfom the following steps:
1. Calculate the derivative f'.
2. Solve f'(x) =0.
3. Determine if the x value found in 2 is a minimum or a maximum using f''.
Recall the following properties of derivatives
where c is a constant.
where f,g are differentiable.
where c is a constant.
(chain rule)
Case 1: f(x) = 2+3x+3.
Using the properties from above, we have
1.
2. The equation f'(x)=0 where f'(x) = 3 has no solution.
3. Based on the previous result, f has no maximum nor minimum.
Case 2:
1.
2. We have the equation
which is equivalent to
Recall that the cosine function only takes values in the set [-1,1]. So, this equation has no solution.
3. Based on the previous result, f has no maximum nor minimum.