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- Determine the critical point of the function and use the critical studied to classify it(s) as a maximum, minimum or saddle point. f(x,y)=-5x²+4xy-y²+16x+10Find the absolute extrema of the function on the closed interval. g(x) = 6x2/ x − 2 , [−2, 1] minimum (x, y) = (smaller x-value)minimum (x, y) = (larger x-value)maximum (x, y) =Find the exact location of all the relative and absolute extrema of the function. (Order your answers from smallest to largest x.) g(x) = 4x3 − 12x + 6 with domain [−2, 2] g has ---Select--- at (x, y) = . g has ---Select--- at (x, y) = . g has ---Select--- at (x, y) = . g has ---Select--- at (x, y) = .
- Considere the function f(x)=3x4-5x3+2. Find all critical points of the function. b. Using the first derivativw test (showing your work), determine and indicate i. Interval(s) of increase ii. Interval(s) of decrease c. Based of the work above, characterize each of your critical points as a local maximum, local minimum, or neither.Find the absolute extrema of the function on the closed interval. g(x) = 4x2 − 16x, [0, 5] minimum (x,y) = maximum (x,y) =Sketch the graph of a di£erentiable function y = ƒ(x) that hasa. a local minimum at (1, 1) and a local maximum at (3, 3);b. a local maximum at (1, 1) and a local minimum at (3, 3);c. local maxima at (1, 1) and (3, 3);d. local minima at (1, 1) and (3, 3).
- A right circular cone is held with its apex facing downwards and is inscribed in a sphere with fixed radius k cm within the interval 1 cm ≤ k ≤ 5 cm and the distance from the base of the cone to the center of the sphere is x cm. a. Evaluate the maximum volume of the cone by assigning k to a value within the given interval. Discuss the difference of use between the first derivative test and second derivative test to optimize the volume. b.Water is poured into the cone at a rate of 10 m3s−1. Find the rate at which the water level is rising when the depth of the water is (x + k) − 3.Find all critical points of f(x, y) = sin(x) sin(y). Classify each one as a local maximum, local minimum, or saddle point. (To check your answer, have a look at a contour plot of this function.)Consider the function f(x)=x* (sq. root 3)-2sinx on [0,pi] Find all critical points of the function. b. Using the first derivativw test (showing your work), determine and indicate i. Interval(s) of increase ii. Interval(s) of decrease (For purpose of estimation, use sq. root 3 as approm. equals 1.7 c. Based of the work above, characterize each of your critical points as a local maximum, local minimum, or neither. (for extra credit: Find both coordinates, in exact form, of the absolute extrema. Credit here only if you show work). help
- Sketch the graph of a differentiable function y = ƒ(x) that has local maxima at (1, 1) and (3, 3);An open–top box with a square base is to be constructed from two materials, one for the bottom and one for the sides. The volume of the box must be 18 cubic feet. The cost of the material for the bottom is 4 pesos per square foot, and the cost of the material for the sides is 3 pesos per square foot. Find the dimensions of the box such that the cost is at its minimum. Find the domain of the function, the critical points and include the second derivative test.Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point . Explain your reasoning. Then use the Second Derivative Test to confirm your predictions . 3.f(x,y)=4+x3+y3-3xy