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- f(x1,x2) 4 2 0 -2 -4 −1 2 0 1 1 0 2 -1 X1 3-2 x2 Figure 2: Graph of the function f(x1, x2) = 2x1 = x²-x+3 Consider the following function: f(x1, x2) = 2x1 - x-x+3. - For your visual convenience, a portion of the function is depicted in Figure 2. (a) Compute the gradient vector of f. (b) Compute the Hessian matrix of f. (c) Determine all points that yield stationary values of f. (d) Use the second-order conditions to determine whether each of the stationary val- ues of f is a maximum, a minimum, or neither.6). Compute the directional derivative of function f(x,y) = xy³ + 2.x²y at the point (2, 1) in the direction (,, ).Find the derivative of f(x)=32x3+14x2+2x+1 where x=2 . Select one: a. 24 b. 21 c. 6 d. 18
- For what value(s) of x is the function f(x) NOT continuous? A x = 0 and x = 3 C. x = -3 only Select one: = 4 2x² — 6x B. x = 3 only D. x = 3 and x = 4The Augsburg math club paid $200 for fancy souvenir water bottles that they are going to sell next year for $6 each at the Taste of Augsburg during Homecoming Weekend. Define two variables that relate quantities in this story. Which one is the input of this function and which one is the output? Write an equation for your function, and briefly explain why your function is valid.4. Suppose that Kyle's value function for cookies can be described as follows: x ≥0 x < 0 a) What does λ denote? V(x) x b) Assuming 2 = 2 fill in the following table: -5 -4 -3 -2 -1 0 1 2 3 4 = 5 s x, Ιλπ, V(x) c) Plot Kyle's value function in a diagram and comment on its shape.
- Quadratic Functions #2 (in image)4. Find the maximum values of the following functions and show whether they are maximized at their maximum values. (a) y= f(x)=10x – x² (b) z=10x+10y-xy-x² - y²Compute the first and second derivatives of the following functions of x. In each case, determine whether the function has any local extremes (maxima or minima) on the domain of nonnegative real numbers – i.e. for x E [0, 0). a. f(x) = 4 + 3x? b. f(x) = x (4x – 3) (9 - 5x) с. f(x) — x2 d. f(x) = In(x) -
- Find the inverse of the given function. f(x) = 5x + 4 .... 11 The inverse of the function is f1 (x) = -X. 5 (Simplify your answer. Use integers or fractions for any numbers in the expression.)Is the function v(x1,x2)=x21x32 a monotonic transformation of u(x1,x2) ? Select one: a. No b. Yes5. Marginal versus average effects Consider the nonlinear function shown by the blue curve on the following graph. The grey line is tangent to this curve at point A. Use the grey line (star point) to answer the following question. (Note: You will not be graded on your placement of this line.) > 10 9 8 7 A 6 4 3 2 1 Tangent Line f(x) 0+ + + + + + 0 1 2 3 4 5 6 7 8 9 10 X The marginal effect of X on Y at point A is The average effect of X on Y at point A is True or False: For this graph, the marginal effect decreases as X increases. True False Scratch Line (?)
