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If f is continuous on the open interval (a, b), then (a) f(x) obtains a maximum value f(c) and a ...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) +...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
f(x) is continuous on (-00,00) and has critical numbers at x = a, b, c, and...
f(x) is continuous on (-00,00) and has critical numbers at x = a, b, c, and d Use the sign chart for f'(x) to determine whether f has a local maximum, a local minimum, or neither at each critical number f'(x) + + + 0 + + + ND ND + + + 0 - h c d Does f(x) have a local minimum, a local maximum, or no local extremum atx=a? Choose the correct answer below. © A. no...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d)...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
Find the largest open interval on which the graph of the function f (x) = x4...
Find the largest open interval on which the graph of the function f (x) = x4 +6x3 x is concave down Use interval notation, with no spaces in between numbers and brackets. For example: (3,8) Answer: Which of the following statements are true about the function below on the interval [a,b]? AA The derivative is 0 at two values of x both being local maxima. The derivative is 0 at two values of x, one on the interval [a,b] while...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a,...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
A function is continuous on the closed, bounded interval [-2, 1] , and differentiable on the...
A function is continuous on the closed, bounded interval [-2, 1] , and differentiable on the open interval (-2,1)Given that f(-2) = 1 , and that the derivative of f is between –5 and —2 throughout the open interval, what is the least possible value of f (1) ? What is the greatest possible value of f(1) ? HINT: Since The greatest possible value of f(1) is f(1) – f(-2) f(1) – f(-2) ^ = f'(c) for some c€ (-2,...
Consider the function f(x) = 14x2 + 200 on the open interval (0,00). (1) Find the...
Consider the function f(x) = 14x2 + 200 on the open interval (0,00). (1) Find the critical value(s) off on the open interval (0, 0). If more than one, then list them separated by commas. Critical value(s) = Preview (2) Find f''(x) = Preview (3) Looking at f''(x) we can conclude the following: f''(x) > 0 for all 3 on the interval (0,0) and thus we have an absolute maximum at the critical value f''(x) < 0 for all x...
Theorem 2.3.1 If f is continuous on an open rectangle (a) that contains (xo yo) then the initial value problem f(a, y), y(o)yo has at least one solution on some open subinterval of (a, b) that co...
Theorem 2.3.1 If f is continuous on an open rectangle (a) that contains (xo yo) then the initial value problem f(a, y), y(o)yo has at least one solution on some open subinterval of (a, b) that contains ro (b) If both fand fy are continuous on R then (2.3.1) has a unique solution on some open subinterval of (a, b) that contains ro. In Exercises 1-13 find all (xo, Vo) for which Theorem 2.3.1 implies that the initial value problem...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
(5 points) A continuous function f, defined for all x, has the following properties: 1. f...
(5 points) A continuous function f, defined for all x, has the following properties: 1. f is decreasing 2. f is concave up 3. f(26) = -5 4. f'(26) = - Sketch a possible graph for f, and use it to answer the following questions about f. A. For each of the following intervals, what is the minimum and maximum number of zeros f could have in the interval? (Note that if there must be exactly N zeros in an...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
f(x) is continuous on (-00,00) and has critical numbers at x = a, b, c, and d Use the sign chart for f'(x) to determine whether f has a local maximum, a local minimum, or neither at each critical number f'(x) + + + 0 + + + ND ND + + + 0 - h c d Does f(x) have a local minimum, a local maximum, or no local extremum atx=a? Choose the correct answer below. © A. no...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
Find the largest open interval on which the graph of the function f (x) = x4 +6x3 x is concave down Use interval notation, with no spaces in between numbers and brackets. For example: (3,8) Answer: Which of the following statements are true about the function below on the interval [a,b]? AA The derivative is 0 at two values of x both being local maxima. The derivative is 0 at two values of x, one on the interval [a,b] while...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
A function is continuous on the closed, bounded interval [-2, 1] , and differentiable on the open interval (-2,1)Given that f(-2) = 1 , and that the derivative of f is between –5 and —2 throughout the open interval, what is the least possible value of f (1) ? What is the greatest possible value of f(1) ? HINT: Since The greatest possible value of f(1) is f(1) – f(-2) f(1) – f(-2) ^ = f'(c) for some c€ (-2,...
Consider the function f(x) = 14x2 + 200 on the open interval (0,00). (1) Find the critical value(s) off on the open interval (0, 0). If more than one, then list them separated by commas. Critical value(s) = Preview (2) Find f''(x) = Preview (3) Looking at f''(x) we can conclude the following: f''(x) > 0 for all 3 on the interval (0,0) and thus we have an absolute maximum at the critical value f''(x) < 0 for all x...
Theorem 2.3.1 If f is continuous on an open rectangle (a) that contains (xo yo) then the initial value problem f(a, y), y(o)yo has at least one solution on some open subinterval of (a, b) that contains ro (b) If both fand fy are continuous on R then (2.3.1) has a unique solution on some open subinterval of (a, b) that contains ro. In Exercises 1-13 find all (xo, Vo) for which Theorem 2.3.1 implies that the initial value problem...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
(5 points) A continuous function f, defined for all x, has the following properties: 1. f is decreasing 2. f is concave up 3. f(26) = -5 4. f'(26) = - Sketch a possible graph for f, and use it to answer the following questions about f. A. For each of the following intervals, what is the minimum and maximum number of zeros f could have in the interval? (Note that if there must be exactly N zeros in an...
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