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# Activity 10 1. Consider the function f (x) = 2ex x (a) Find any critical points of f . (b) Explain why x = 0 is not a critical point of f (x). Hint:...

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Activity 10 1. Consider the function f ( x ) = 2 e - x x (a) Find any critical points of f . (b) Explain why x = 0 is not a critical point of f ( x ) . Hint: Think about the domain of f 2. Let y = x (2 x +1) . Find the equation of the line tangent to the graph of y at the value x = 1 . You will need to use logarithmic diﬀerentiation. 3. Suppose a ﬁrm has a revenue function R ( x ) where R is the monthly revenue for the ﬁrm and x is the amount the ﬁrm spends on advertising each month. (a) Explain the meaning of R 0 ( x ) in the context of this problem. In other words describe or explain what R 0 ( x ) tells you about the relationship between the ﬁrm’s monthly revenue and the ﬁrm’s monthly spending on advertising. (b) Do you expect R 0 ( x ) to always be positive, negative, or zero (for positive x )? Why or why not? (c) If you had an algebraic expression for R ( x ) what would be the steps to ﬁnd the point of diminishing returns for ad expenditures? 4. How much would you need to set aside now, at an interest rate of 5.5%, compounded continuously, in order have \$50,000 in 6 years. (We are assuming you are just making a one time deposit- as opposed to putting away money every month- for example) (This is a present value problem) 1

1:
f(x) =
f’(x) = 2 [ for maxima or minima f’(x) =0
so, 2 [
or2 [
or, x= 1 [as, =0=0 ANS:
a&gt; Critical point of x is x= 1 b&gt; x=0 can not be critical point as the
function becomes undefined...

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