Question
Q1)
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Q7)
If U and V are positive constants, find all critical points of the function. (Enter your answers as a comma-separated list.) F(t) = Ue t + Ve -t t = Find all critical points of f(x) = 20 x 2 e - 0.3x and classify them as local minima of ,maxima. x = (smaller value) This critical point is a . x = (larger value) This critical point is a . Consider f(x) = x 8 (7 - x) 9. Find f '(x) and simplify your answer. f '(x) = Find all critical points of f(x) and then use the first derivation test to determine local maxima and minima. f(x) has a at x = (smallest value) Local Maximum OR Minimum OR Neither f(x) has a at x = Local Maximum OR Minimum OR Neither f(x) has a at x = (largest value) Local Maximum OR Minimum OR Neither Find the inflection points of f(t) = t 4 + t 3 - 18t 2 +8. Give exact answer. t = (smallest value) t = (largest value) Sketch a possible of y = f(x), using the given information about derivation y' = f'(x) and y" = f"(x). Assume that the function is defined and continuous for all real x. Let f be a function f(x) > 0 for all x. Let g = 1/f. If f is increasing in an interval around x 0, what about g? g is increasing in an interval x 0. g is zero in an interval around x 0. g is decreasing in an interval around x 0. We cannot tell whether g is increasing or decreasing in an interval around x 0. g is constant in an interval around x 0. If f has a local maximum at x 1, what about g? g has a local maximum at x 1. g has both a local minimum and a local maximum at x 1. We cannot tell whether g has a local minimum or a local maximum at x 1. g has an inflection point at x 1. g has a local minimum at x 1. If f is a concave down at x 2, what about g? g is increasing at x 2. g is flat at x 2. We cannot tell whether g is concave up of concave down at x 2. g is concave down at x 2. g is concave up at x 2. You are given the graph of the second derivation function f" below. Which of the given have x-values that are inflection points of the function f? (Select all the apply.) A B C D E F G H I none of the above
Explanation / Answer
Q1)
f ' (t)=0
==>ue^t -ve^-t=0
==>ue^t=ve^-t
==>e^2t=v/u
==>t=0.5 ln(v/u)
Q2)f ' (x)=0
==>40xe^-0.3x -6x^2e^-0.3x=0
==>xe^-0.3x *(40-6x)=0
==>x=0, x=20/3
x=0==>f(0)=0------------>local minimum
x=20/3 ==>f(20/3)>0--->local maximum
Q3)F'(X)=0
==>8x^7 (7-x)^9 +-9x^8 (7-x)^8 =0
==>x^7 (7-x)^8 (8*(7-x) -9x)=0
==>x^7 (7-x)^8 (56-17x)=0
==>x=56/17 ,0,7
local minimum at x=0
local maximum at x=56/17 or 3.294
neither at x=7
Q4)F"(T)=0
f'(t)=4t^3+3t^2 -36t
f"(t)=12t^2 +6t -36=0
==>2t^2 +t -6=0
==>(t+2)*(2t-3)=0
==>t=-2---->smaller
, t=3/2=1.5-->larger
Q5)option A
Q6)a)g is decreasing in interval around x0
b)local minimum
c)concave up
Q7)D,F as f"(x) =0 at these points
