SHOW ALL YOUR WORK. Indicate clearly the methods you use because you will be gra

Question

SHOW ALL YOUR WORK. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as on the accuracy of your final answers. If you choose to use decimal approximations, your answer should be correct to three decimal places. Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Let f be the function given by f(x) = x^3 - 5x^2 + 3^x + k, where k is a constant. On what intervals is f increasing? On what intervals is the graph of / concave downward? Find the value of k for which f has 11 as its relative minimum.

Explanation / Answer

a) f is increasing when f dash(x) > 0

so f dash (x) = 3x^2 -10x +3 >0

solving the quadratic inequality we get

x>3 or x<1/3 --------ANSWER a

b) f double dash (x) = 6x-10

we have 2 inflection points from part a) 3 and 1/3

take one point <1/3 , one point in between 3 and 1/3 and one point greater than 3

let's I take ( 0) , (2) and (4)

f double dash (0) = -10 <0 ( so concave downwards)

f double dash (2) = 6*2-10 = 2 >0 ( concave upwards)

f double dash (4) = 6*4-10 = 14 >0 (concave upwards)

so interval in which curve is concave downwards is (-infinity , 1/3)

c) we will have minimum at x= 3 , bcause f double dash (3) is >0

so 11 = 3^3-5*3^2+3*3+k, solving

we get k = 20 -----ANSWER

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