How do I approach this?
How do I approach this? *58. a) If g is a given function which is continuous...
1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has a unique and continuous solution φ(t) on a small interval containing to, φ(t) satisfies the initial condition φ(to) = to. 1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has...
Sub-problem 2. Recall the eztreme value theorem: If g(ax) is continuous on (0, 1], then g attains its marimum. In particular it is BOUNDED. L.e. there is a number M20 such that 1. Find a number M so that lg(a)l s M, where g(x)x(1-) (Hint: marimize g 8. Provide an erample of a function that is NOT continuous on [0, 1] but IS bounded. 3. Provide an ezample of a function that is NOT continuous on [0, 1] and that...
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (1 - 1)2 + y2 + 3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (a) Write down a Lagrangian function L(x, y, 1) whose only stationary point (x*, y*, \*) corre- sponds to the point of tangency (r*, y*) between the line...
a. Find the open interval(s) on which the function is increasing and decreasing b. Identify the function's local and absolute extreme values, if any, saying where they occur Determine the open interval(s) of x for which g(x) increases. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. o A. The function is increasing on the open interval(s) Type your answer in interval notation. Use a comma to separate answers as needed.)...
"Functions of two or more variables can behave very differently than functions of only one variable. Here are two examples to explore to illustrate thatclaim.(a) If a function of one variable is continuous on an interval and has only one critical point that is a local minimum, then it must be a global minimum. But thisis not true for a continuous function of two variables. Letf(x,y) = x^2 * (y+1)^3 + y^2Verify that (0,0) is the only critical point and...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
how do i minimize this function. 21L + (2600/L)Set the derivative equal to zero and solve for L. 21 - 2600/L^2 = 0 2600 = 21 L^2 There will be positive and negative solutions for L. You can use the second derivative test or direct calculation to see which gives a minimum.
A continuous function f, defined for all x, has the following properties:1. f is increasing2. f is concave down3. f(5)=24. f'(5)=1/2Sketch 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 aninterval, the minimum and maximum are both N.)minimum maximum- infinity < x _<...
/2 POINTS ZILLDIFFEQMODAP11 4.4.042 MY NOTES ASK YOUR TEACHER Solve the given initial value problem in which the input function of) is discontinuous Hint: Solve the problem on two intervals, and then find a solution so that y and continuous at are 0) = 0, where y" - 2y + 10 (x), YO) - 0, ()-80, OSX lo Need Help? Red Wich t e Tutor
rt) dt, where f is the function whose graph is shown. /, 0 Let g(x)- f(t) 2 (a) At what values of x do the local maximum and minimum values of g occur? Xmin xmin = xmax = Xmax (smaller x-value) (larger x-value) (smaller x-value) (larger x-value) (b) Where does g attain its absolute maximum value? (c) On what interval is g concave downward? (Enter your answer using interval notation.) (d) Sketch the graph of g. 0.5 -0.5 2 46...