Answer the following questions about the function f(x) = (11 - 2x)e^x. Instructi

Question

Answer the following questions about the function f(x) = (11 - 2x)e^x. Instructions: If you are asked to find x- or y-values, enter either a number, a list of numbers separated by commas, or None If there aren't any solutions. Use Interval notation if you are asked to find an Interval or union of intervals, and enter {} If the interval is empty Find the critical numbers of f, where it is Increasing and decreasing, and its local extrema. Critical numbers x = ______ Increasing on the interval ___________ Decreasing on the interval __________ Local maxima x = __________ Local minima x = _____________ Find where f is concave up, concave down, and has inflection points. Concave up on the interval __________ Concave down on the Interval _________ Inflection points x = _________ Find any horizontal and vertical asymptotes of f. Horizontal asymptotes y = ____________ Vertical asymptotes x = _____________ The function f is ____ because _______ for all x in the domain of f, and therefore its graph is symmetric about the ___________ Sketch a graph of the function f without having a graphing calculator do it for you Plot the y-intercept and the x-intercepts, if they are known. Draw dashed lines for horizontal and vertical asymptotes. Plot the points where f has local maxima, local minima, and Inflection points. Use what you know from parts (a) and (b) to sketch the remaining parts of the graph of f. Use any symmetry from part

Explanation / Answer

calculate f'(x) (use product rule)

f'(x) = (-2) e^x + (11-2x) e^x = (9 -2x)e^x

critical numbers => when f'(x) not defined or zero

so x = 9/2 is the critical number

increasing function => f'(x) > 0 => x < 9/2

decreasing function => f'(x) < 0 => x > 9/2

there isonly one critical point and it is a point of maxima .Since f''(9/2) = (7-2*9/2)e^(9/2)< 0

f"(x) = (7-2*x)e^(x) <0 when x > 7/2

and > 0 when x<7/2

so for x<7/2 graph is concave up

and for x>7/2 it is concave down

there is no point of local minima

x= 0 is the hrizontal asymptote

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