answer:Given the function {eq}g(x, y) = (xy)^{frac{1}{3}}{/eq}: a) The function is continuous at (0, 0) because the limit exists and is equal to the function value at that point. As x, y to 0, we have: {eq}lim_{{x,y to 0,0}} g(x, y) = lim_{{x,y to 0,0}} (xy)^{frac{1}{3}} = 0 = g(0,0){/eq}. b) The partial derivatives are: {eq}frac{partial g}{partial x} = frac{1}{3}y^{frac{1}{3}}x^{frac{-2}{3}}{/eq} and {eq}frac{partial g}{partial y} = frac{1}{3}x^{frac{1}{3}}y^{frac{-2}{3}}{/eq} when xy neq 0. c) At (0, 0), both partial derivatives exist and are equal to zero: {eq}frac{partial g}{partial x}Bigg|_{(0,0)} = 0{/eq} and {eq}frac{partial g}{partial y}Bigg|_{(0,0)} = 0{/eq}. d) The partial derivatives are not continuous at (0, 0) because their limits as x, y to 0 do not exist. This is due to the indeterminate form {eq}0^0{/eq}. e) The graph of g(x, y) does not have a tangent plane at (0, 0), as the function is not differentiable at this point. If we consider the line y = mx and evaluate g(x, y) along this line, we find that the directional derivative exists in all directions except along the line x = 0. f) The function g(x, y) is not differentiable at (0, 0), as the partial derivatives are not continuous at that point. This implies that there is no unique tangent plane to the graph at (0, 0).

answer:To multiply a scalar by a matrix, perform the scalar multiplication on each element of the matrix. Here's the step-by-step process: 1. Multiply each element of the first row by the scalar frac{7}{50}: - 4 times frac{7}{50} = frac{28}{50} = frac{14}{25} - 9 times frac{7}{50} = frac{63}{50} - -5 times frac{7}{50} = frac{-35}{50} = frac{-7}{10} 2. Multiply each element of the second row by the scalar frac{7}{50}: - -4 times frac{7}{50} = frac{-28}{50} = frac{-14}{25} - -7 times frac{7}{50} = frac{-49}{50} - 4 times frac{7}{50} = frac{28}{50} = frac{14}{25} The result is: left( begin{array}{ccc} frac{14}{25} & frac{63}{50} & frac{-7}{10} frac{-14}{25} & frac{-49}{50} & frac{14}{25} end{array} right)

answer:To find the real solutions, we need to consider two cases because the absolute value can be positive or negative: 1. When -24x - 16 is non-negative (geq 0): -24x - 16 = 21 Solving for x gives us: -24x = 21 + 16 -24x = 37 x = -frac{37}{24} 2. When -24x - 16 is negative (< 0): -24x - 16 = -21 Solving for x gives us: -24x = -21 - 16 -24x = -37 x = frac{-37}{-24} x = frac{37}{24} However, since x would be in the interval where the expression inside the absolute value is negative, this solution is not valid for the original equation. Therefore, the real solutions to the equation are: x = -frac{37}{24}

answer:Given that, Mean, {eq}mu = 250{/eq} Standard deviation, {eq}sigma = 80{/eq} The required probability is {eq}P(X > 350){/eq}. Now, {eq}P(X > 350) = P(dfrac{X - mu}{sigma} > dfrac{350-mu}{sigma}) P(X > 350) = P(Z > dfrac{350-250}{80}) P(X > 350) = P(Z > 1.25){/eq} Excel function for the above probability: =1 - NORM.DIST(1.25,0,1,1) {eq}P(X > 350) = 0.1056{/eq}