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f(x)=x4−2x3 (A) Find all critical Values of f. If theee are no critical values, enter - 1000, If there are more than one, enter them separated by commas. Critical value(s) = (B) Use interval notation to indicate where f(z) is increasing. Note: When using interval notation in WeBWork, you use I for ∞, -f for −∞, and U tor the union symbol. If there ave no values that satisfy the required condition, then enter "ff" without the quotation marks. increasing: (C) Use inteval notation to indicate where f(x) is decreasing: Decroating: (i0) Find the x-coordinates of allocal maxina of f, If there are no local macima. enter-1000, if there are more than one, enter them separated by commas. Local maxima at z= (6) Find the z-coordinstes of all local minima of f. If there are no local minima, enter-10od. If there are more than one, enee them separated by comnas. Local mineta at x = (A) Use interval notation to indicalu where f(z) a concave up Poncave up (0) Use interval notation to indicate whies f(x) is concave down. Concave bistr Inflection partish at a =
Please answer the following questions about the function f(x)=x+53x−5​ Instructions: If you are asked to find a function, enter a function. If you are asked to find x - of yvalues, enter ecther a number, a list of numbers separated by cornmas, 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. (a) Calculate the first derivative of f. Find the critical numbers of f. where it is increasing and decreasing, and its local extrema. f′(x)= Critical numbers x= Increasing on the interval Decreasing on the interval Local maxima x= Local minima z = (b) Calculate the second derivative of f. Find where f is concave up, concave dewn, and has inflection points. f′′(x)= Concave up on the interval Concave down on the interval tiflection points z= (c) Find ary horzontal and vertical agymptotes of f. Horitiontal asyrnptotes y = Vertical asymptotes ₹= (G) The function f in because for al x in the domain of f, and therefore its graph is symmetrie about the (e) Sketch a graph of the tunction f without having a ghaphing calculator do it for you. Plot the y-intercept and the x-thtercepts, if they an known, Draw dashed ines for horizontal and vertical asymptotes. Pot the points whore f has local makima, local minima, and infection points. Use what you know trom parts (a) and (o) to sketch the romaining parts of the graph of f. Use any symmetry tom part (क) to your advantage. Sketching graphs is an important shal that takes practice, and you may be asked to do it on quetzes or exams:
Please answer the following questions about the function f(x)=x2−94x​ Instructions: If you are asked to find a function, enter a function. 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 ary solutions. Use interval notation if you are asked to find an interval or union of intervals, and enter (\} if the interval is empty. (6) Calculate the first derivative of f. Find the critical numbers of f; where it is increasing and decreasing, and its local extrema. f′(z)= Critical rumbers x= Increasing co the interval Decreasing on the interval Local maxima z= Local minima x= (b) Calculase the second dertvative of f. Find where f is concave up, ooncawe down, and has intlection points. f′′(x)= Concave up on the interval Concive down on the interval intlection points x=
f(x)=2x2ln(x),x>0 (A) List all critical numbers of f. If there are no critical values, enter 'NONE'. critical numbers = (B) Use interval notation to indicate where f(x) is increasing. Note: Use 'INF' for ∞, '-INF' for −∞, and use ' U ' for the union symbol. Increasing: (C) Use interval notation to indicate where f(x) is decreasing. Decreasing: (D) List the x-coordinates of all local maxima of f. If there are no local maxima, enter 'NONE'. x values of local maxima = (E) List the x-coordinates of all local minima of f. If there are no local minima, enter 'NONE'. x values of local minima = (F) Use interval notation to indicate where f(x) is concave up. Concave up: (G) Use interval notation to indicate where f(x) is concave down. Concave down: (H) List the x values of all inflection points of f. If there are no inflection points, enter 'NONE:. ± values of inflection points = (i) Use all of the preceding information to sketch a graph of f. When you're finished, enter a "1" in the box below. Graph Complete:
Consider the function f(x)=4+exex​ Then f′(x)=(4+ex)24ex​ The following questions ask for endpoints of intervals of increase or decrease for the function f(x). Write INF for ∞, MINF for −∞, and NA (ie. not applicable) if there are no intervals of that type. The interval of increase for f(x) is from to The interval of decrease for f(x) is from to f(x) has a local minimum at (Put NA if none.) f(x) has a local maximum at (Put Na if none.) Then f′′(x)= The following questions ask for endpoints of intervals of upward and downward concavity for the function f(x). Write INF for ∞, MINF for −∞, and put NA if there are no intervals of that type. The interval of upward concavity for f(x) is from to The interval of downward concavity for f(x) is from to f(x) has an inflection value, x= (Put NA if none)
The picture below shows the graph y=f′(x) of the derivative of a function y=f(z). For each of the labelled points on the graph, classify the corresponding point on the graph of y=f(x) as on of the following: Max, MiN, NFL, INT (short for maximum, minimum, inflection point, x-intercept) A: MPN B: NFL C: MAX D. BFL E. MIN For each of the follewing intervals, classify whether the graph of y=f(z) is iNC of DEC over that interval (thort for increasing or decreasingl. (−∞,A)DEC (A,B)INCINC​ (B,C)INC (C,D)DEC (E,∞)DC For each of the following intervala, classhy whether the graph of y=f ) is cU or cD over that interval (ahort for concave Lp or concave downh) (−∞,A)CU (A,B)(B,C)​CUCD​ (B,C)CD (C,D)CD (D,b)CU (E,∞)CU
For the function f given above, determine whether the following conditions are true. Input T if the condition is ture, otherwise input F. (a) f′(x)<0 if 0<x<2 (b) f′(x)>0 if x>2; (c) f′′(x)<0 if 0≤x<1; (d) f′′(x)>0 if 1<x<4. (e) f′′(x)<0 if x>4; (f) Two inflection points of f(x) are, the smaller one is x= and the other is x=