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- Find the slope of the tangent line to the parabola y=4x-x^2 at the point (1,3) 1) using the definition: m=lim->a f(x)-f(a)/x-a 2)using the equation: m=lim->0 f(a+h)-f(a)/hDetermine value for c so that lim f(x) x--> 3 exists f(x)= 1/3x+c For x<3 -x+8 For x>3For every solution y(x) of the equation y''+ 2y'+ 5y = 0 it is true that limx→∞ y(x) = 0. ?
- Show that the function f(x,y)=8x^2 y subject to 3x−y=9 does not have an absolute minimum or maximum. (Hint: Solve the constraint for y and substitute into f.) Solve the constraint for y. y = ? Substitute into f. f(x,y)= ? Determine the behavior of f as x approaches −∞. limx→−∞f(x,y)= ? Determine the behavior of f as x approaches ∞. limx→∞f(x,y)= ? Does this show that f does not have an absolute maximum or minimum? 1. No 2. YesSuppose that a function ƒ satisfies the following two conditionsfor all real values of x and y:i) ƒ(x + y) = ƒ(x) . ƒ(y).ii) ƒ(x) = 1 + xg(x), where limx→0 g(x) = 1.Show that the derivative ƒ'(x) exists at every value of x and thatƒ'(x) = ƒ(x).Suppose g(x)<h(x)forx∈(0,1)∪(1,2) both limx→1g(x)andlimx→1h(x) exist. Must limx→1g(x)<limx→1h(x)? true or false