The function f(x, y) = x²y + xy2 – axy (a>0) has a a (A) a maximum at 3'3 a a (B) a minimum at 3 3 а а (C) a saddle point at 3 3 a a (D) neither maximum nor minimum at 3'3
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- 1. Examine the function F(x, y) = 5x+ 2xy+ 5y++ 96x+ 288y for relative extrema. a. relative minimum at 4. -28) b. relative maximum at (4,-28) (4, 28) C. relative minimum at d. relative maximum at (-4, 28) e. no relative extremaFind the maximum point of the curve y=---- -2x+1 2 A. 3 (2-) B. C. D.What are the coordinates of the points in the graph of y2-x²+6x – 13=0 in which the tangent line is horizontal? (A (4, – 5). (4,5) (2, – V5).(2. /5) O (1.-2/2).(1.22) D (3, – 2), (3,2)
- Use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. f(x, у) %3D —х2 - бу? + 12х — 36у — 87 (х, у, 2) %3D maximumFind t for the following terminal points: (a) P(0, 1), t = (b) P(-, 부), t%3D 2 2 (c) P(-1,0), t = (4) P(-, -), t =| (е) Р(0, —1), t %3 | () P(, -),t =Find the parabola y = ax2 + bx + c that passes through (2,1) and be tangent to the line y = 2x + 4 at (1,6)
- A geoscientist is logging the profile of the seabed to discover the hydrocarbon. The map can be represented by General Mathematical Model M(x, y) = —D Ах" + Ву" — Сх — Dy + E - i. Propose the values for m, n, A, B, C, D, and E such that it will give three extreme seabed profiles M(x, y). Verify it with the derivative test. i. Referring to the directional derivatives, find the extrema and suggest a best place to land an oil rig. Explain your reasons.A rectangular plate with an area of 36 cm2 is to be produced on a lathe in such a way that the perimeter, P, of the plate, is a minimum. (a) If the length of the plate is x cm and the breadth is y cm, draw a diagram and show that the perimeter, P, can be expressed in terms of x only as: P = 2x + 72x-' cm. d²P dP and dx (b) Find and use these derivatives to determine the dx? dimensions of the component for which the perimeter is a minimum. Calculate the minimum perimeter.Which of the following straight lines are tangent to the curve y = x³ + 4x2 + 7x and passes through the point (0 , 0)? (1) y = 7x (II) y = - 11x %3D (III) y = 11x (IV) 3x + y = 0 (V) 3x - y = 0 Lines I, and V O Lines I , and II Lines II , and IV O Lines III , and IV Only Line I
- Use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. fx, у) %3D 5x2 - 4ху + у2 - бу (х, у, 2) 3D (| 10, — 4,2 minimumThe total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is approximated by R(x,y) = 16+329x+275y-8x²-6y² - 11xy. Find the number of each that should be sold to produce maximum revenue. Find the maximum revenue. Find the derivatives Rxx, Ryy, and Rxy- Rxx = -16, Ryy = -12, Rxy = -11 Selling spas and solar heaters gives the maximum revenue of $ (Simplify your answers.)The point P(7, -2) lies on the curve y (a) If Q is the point (x, (x₁ (i) 6.9 mpQ (ii) 6.99 mpQ (iii) 6.999 тра (iv) 6.9999 MpQ (v) 7.1 тра (vi) 7.01 mpQ = m = = (vii) 7.001 MpQ (viii) 7.0001 mpQ 2 6- X 2 6- X find the slope of the secant line PQ (correct to six decimal places) for the following values of x. (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(7, -2). (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(7, -2).