2018 international practice exam ab mcq, part b - calculator required let f be a twice - differentiable…

2018 international practice exam ab mcq, part b - calculator required let f be a twice - differentiable function such that f(x) < 0 for all x. the graph of y = s(x) is the secant line passing through the points (3, f(3)) and (5, f(5)). the graph of y = t(x) is the line tangent to the graph of f at x = 4. 13 mark for review which of the following is true? a f(4.2) < s(4.2) < t(4.2) b f(4.2) < t(4.2) < s(4.2) c s(4.2) < f(4.2) < t(4.2) d t(4.2) < f(4.2) < s(4.2)

2018 international practice exam ab mcq, part b - calculator required let f be a twice - differentiable function such that f(x) < 0 for all x. the graph of y = s(x) is the secant line passing through the points (3, f(3)) and (5, f(5)). the graph of y = t(x) is the line tangent to the graph of f at x = 4. 13 mark for review which of the following is true? a f(4.2) < s(4.2) < t(4.2) b f(4.2) < t(4.2) < s(4.2) c s(4.2) < f(4.2) < t(4.2) d t(4.2) < f(4.2) < s(4.2)

Answer

Explanation:

Step1: Analyze the function property

Since (f''(x)<0) for all (x), the function (y = f(x)) is concave - down.

Step2: Recall the relationship between secant and tangent lines for concave - down functions

For a concave - down function, the tangent line at a point (x = a) lies above the graph of the function, and the secant line between two points ((x_1,f(x_1))) and ((x_2,f(x_2))) lies above the graph of the function for (x\in(x_1,x_2)) and the tangent line at (x=\frac{x_1 + x_2}{2}) lies above the secant line. Here (x_1 = 3), (x_2=5), and we are considering (x = 4.2). The secant line (y = S(x)) passes through ((3,f(3))) and ((5,f(5))), and the tangent line (y = T(x)) is at (x = 4). For a concave - down function (f(x)), we have (f(x)<S(x)) for (x\in(3,5)) and (S(x)<T(x)) (because of the concavity property). So (f(4.2)<S(4.2)<T(4.2)).

Answer:

A. (f(4.2)<S(4.2)<T(4.2))