5. The function g(x) is continuous on the closed interval [-2, 0] and twice differentiable on the openinterval (-2, 0). If g'(-1)=-2 and g"(x) > 0 on the open interval (-2, 0), which of the following could bea table of values of g?X-2-10A.g(x)531X-2-10B.g(x)541Y-2-10C.g(x)569D.X-2-10g(x)579X1-2-10E.g(x)7.543.5

Given: 1. gx is a continuous function in the interval x∈-2,0. 2. gx is double differentiable in the interval x∈-2,0. 3. g'-1 = -24. g''x > 0-------------------------Aim: To determine the possible value table for x Vs gx from the following tables. -------------------------------------------------------------For interval x∈-2,0: gx is double differentiable means, its quadratic in nature. gx = ax2 + bx+c-----------------------------g'-1 = -2 It means the slope of gx at x=-1 is negative with magnitude 2. This implies that across x=-1, the gx is decreasing by '2' with unit change in the value of x.Using central difference approximation: g'-1=g0-g-22g0-g-22=-2----------------------------g''x >0 for x∈0,2. This implies that the gx curve is concave upward. Or the slope of the curve is continuously increasing. ---------------------------- Curve A Curve B The above curve is continuous and quadratic in nature. Concavity is upwards. The above curve is continuous and quadratic in nature. Concavity is upwards. The slope is negative at x=-1 or g'(-1) <0. The slope is positive at x=-1 or g'(-1) > 0. Hence, curve A is a possible nature of the given function g(x). The curve A is a possible nature of the given function g(x). The implies that g(x) curve is decreasing in the given interval (otherwise g(x) can not have negative slope at x=-1). Hence, table C & table D are incorrect choices for g(x). ------------------------------------------------------------------------ g''(x) > 0 implies, the fall of the curve decreases with 'x'. Hence, g'(-2) > g'(-1) > g'(0) . It means the fall in the value of g(x) from x=-2 to x=-1 should be more than the fall from x=-1 to x=0. Table A, the fall is constant, hence it is a linear curve. hence it not g(x) curve. Table B, the fall increases with 'x', means g''(x)<0, so it not g(x) curve. In table 'E' , the fall is initially more and then decreases, hence Table E is a correct choice for x Vs g(x).