What does it mean to "give the optimum value of the indicated expression"?
Find the two nonnegative numbers x and y that satisfy these requirements. Give the optimum value of the indicated expression. The sum of x and y is 140 and the sum of the squares of x and y is minimized. Work: x+y=140 y=140-x f(x)=x^2+y^2 =x^2+(140-x)^2 f'(x)=2x+2(140-x)(-1) 0=... x=70 f''(x)=4 f"(70)=4 rel min concave up x+y=140 70+y=140 y=70 x=70, y=70 But what do they mean by "optimum value of the indicated expression"?****
Find the point on the line -5x+4y+4=0 which is closest to the point (5,2)?
Solving the given equation for y yields: -5x + 4y + 4 = 0 ==> y = (5/4)x - 1. Then, the corresponding point on the line of -5x + 4y + 4 will be [x, (5/4)x - 1]. The distance between [x, (5/4)x - 1] and (5, 2) is: D = √{[(5/4)x - 1 - 2]^2 + (x - 5)^2} = √[(41/16)x^2 - (35/2)x + 34]. Then, the x-coordinate of the vertex of (41/16)x^2 - (35/2)x + 34 is: -b/(2a) = -(35/2)/[2(41/16)] = -140/41. Finally, the corresponding y-coordinate is (5/4)(-140/41) - 1 = -216/41. Therefore, the point on the line -5x + 4y + 4 = 0 that is closest to (5, 2) is (-140/41, -216/41). I hope this helps!