TRUE/FALSE If f(x) has a minimum at x=a, then there exists an ε, such that f(x)...
TRUE/FALSE
If f(x) has a minimum at x=a, then there exists an ε, such that f(x) > f(a) for every x in (a- ε, a+ ε).
If x=c is an inflection point for f, then f(c) must be a local maximum or local minimum for f.
f(x) = ax2 +bx +c, (with a ≠ 0), can have only one critical point.
Second Shape Theorem includes the converse of First Shape Theorem.
If f’(x)=g’(x) then f(x)=g(x) +c, where c is a constant.
Homework Answers
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TRUE/FALSE
If f(x) has a minimum at x=a, then there exists an ε, such that
f(x)...
Consider the following graph of f(x) on the closed interval (0,5): 5 4 3 2 1...
Consider the following graph of f(x) on the closed interval (0,5): 5 4 3 2 1 0 -1 0 1 2 3 5 6 (If the picture doesn't load, click here 95graph2) Use the graph of f(x) to answer the following: (a) On what interval(s) is f(x) increasing? (b) On what interval(s) is f(x) decreasing? (c) On what interval(s) is f(x) concave up? (d) On what interval(s) is f(x) concave down? (e) Where are the inflection points (both x and...
True or False Evaluate each of the statements below as true or false.
True or False Evaluate each of the statements below as true or false. a) A critical number of a function f(x) can only be defined as a number c in the domain of f(x) where f'(c) = 0). b) If f(x) is a continuous function on [a,b], then an absolute maximum ſ(c) and an absolute minimum f(d) for some c, d in [a, b] are guaranteed to exist. c) If f'(c) = 0, then f has a local maximum or a local minimum at...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), the...
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
By examining the Hessian matrix, show that if f(x, y, z) has a local minimum at...
By examining the Hessian matrix, show that if f(x, y, z) has a local minimum at (x0, y0, z0), then g(x, y, z) = −f(x, y, z) must have a local maximum at that point. Likewise, show that if f has a local maximum, then g must have a local minimum at that point.
What does the Second Derivative Test guarantee about the point x=2 of the function f(x) =...
What does the Second Derivative Test guarantee about the point x=2 of the function f(x) = .0001(x - 2)4? The point x=2 is a local maximum The point x=2 is a local minimum. The point x=2 is an inflection point. The point x=2 is not a critical point. The Second Derivative Test does not apply to x=2.
1. Suppose that f(x) has a critical number at x=c, and f′′(c)=−10 By the Second Derivative...
1. Suppose that f(x) has a critical number at x=c, and f′′(c)=−10 By the Second Derivative Test, we conclude A. the test is inconclusive. B. x=c is an inflection point C. x=c is a local (relative) minimum D. x=c is a local (relative) maximum E. x=c is an absolute minimum Question 4 of 10 3 Points What follows is a numeric fill in the blank question with 2 blanks. Find the absolute maximum and minimum value of the function f(x)=0.5x^4+(4/3)x^3−3x^2+4...
a) Verify the Rolle's theorem for the function f(x) = -1 x +x-6 over the interval...
a) Verify the Rolle's theorem for the function f(x) = -1 x +x-6 over the interval (-3, 2] 3-X b) Find the absolute maximum and minimum values of function f(x)= (1+x?)Ě over the interval [-1,1] c) Find the following for the function f(x) = 2x – 3x – 12x +8 i) Intervals where f(x) is increasing and decreasing. ii) Local minimum and local maximum of f(x) iii) Intervals where f(x) is concave up and concave down. iv) Inflection point(s). v)...
7. [23] Given the following function:: f(x)-x-4x +6 (a) Find all of the critical points of...
7. [23] Given the following function:: f(x)-x-4x +6 (a) Find all of the critical points of this function. Show your work. (b) Characterize each of the critical points as a local maximum, a local minimum or neither. Show your work. (c) Find all of the inflection points of this function (verify that it/they are indeed inflection points). (d) On what interval(s) is this function both decreasing and concave down? on the interval -15xs1. Show (e)Find the global maximum and minimum...
17. Given the following function and its first and second derivative: 20-2 6-43 f'(x)= f"(x) =...
17. Given the following function and its first and second derivative: 20-2 6-43 f'(x)= f"(x) = [2 pts] 1) Find the horizontal and vertical asymptotes of f(x), if any. f(x)=x-2x=1 نر [2 pts) ii) Find all critical numbers. Note: NOT a point, just critical numbers only. [5 pts) iii) Find the intervals of increasing and decreasing then finding all local maximum minimum values. [5 pts] Find the intervals of concave upward and concave downward. [2 pts) Find inflection point, if...
Consider the following graph of f(x) on the closed interval (0,5): 5 4 3 2 1 0 -1 0 1 2 3 5 6 (If the picture doesn't load, click here 95graph2) Use the graph of f(x) to answer the following: (a) On what interval(s) is f(x) increasing? (b) On what interval(s) is f(x) decreasing? (c) On what interval(s) is f(x) concave up? (d) On what interval(s) is f(x) concave down? (e) Where are the inflection points (both x and...
True or False Evaluate each of the statements below as true or false. a) A critical number of a function f(x) can only be defined as a number c in the domain of f(x) where f'(c) = 0). b) If f(x) is a continuous function on [a,b], then an absolute maximum ſ(c) and an absolute minimum f(d) for some c, d in [a, b] are guaranteed to exist. c) If f'(c) = 0, then f has a local maximum or a local minimum at...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
What does the Second Derivative Test guarantee about the point x=2 of the function f(x) = .0001(x - 2)4? The point x=2 is a local maximum The point x=2 is a local minimum. The point x=2 is an inflection point. The point x=2 is not a critical point. The Second Derivative Test does not apply to x=2.
a) Verify the Rolle's theorem for the function f(x) = -1 x +x-6 over the interval (-3, 2] 3-X b) Find the absolute maximum and minimum values of function f(x)= (1+x?)Ě over the interval [-1,1] c) Find the following for the function f(x) = 2x – 3x – 12x +8 i) Intervals where f(x) is increasing and decreasing. ii) Local minimum and local maximum of f(x) iii) Intervals where f(x) is concave up and concave down. iv) Inflection point(s). v)...
7. [23] Given the following function:: f(x)-x-4x +6 (a) Find all of the critical points of this function. Show your work. (b) Characterize each of the critical points as a local maximum, a local minimum or neither. Show your work. (c) Find all of the inflection points of this function (verify that it/they are indeed inflection points). (d) On what interval(s) is this function both decreasing and concave down? on the interval -15xs1. Show (e)Find the global maximum and minimum...
17. Given the following function and its first and second derivative: 20-2 6-43 f'(x)= f"(x) = [2 pts] 1) Find the horizontal and vertical asymptotes of f(x), if any. f(x)=x-2x=1 نر [2 pts) ii) Find all critical numbers. Note: NOT a point, just critical numbers only. [5 pts) iii) Find the intervals of increasing and decreasing then finding all local maximum minimum values. [5 pts] Find the intervals of concave upward and concave downward. [2 pts) Find inflection point, if...
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