Question

The tangent line to the graph of a twice-differentiable function at a point is used to approximate the value of f(x). Which statement guarantees that the tangent line approximation at x is an underestimate of f(x)?

Option 1: f′′(x)<0
Option 2: f′(x)<0
Option 3: f′′(x)>0
Option 4: f′(x)>0

Answer 1

The statement that guarantees the tangent line approximation at x is an underestimate of f(x) is Option 1: f′′(x)<0.

Step-by-step explanation:

The statement that guarantees the tangent line approximation at x is an underestimate of f(x) is Option 1: f′′(x)<0.

When the second derivative f′′(x) is negative, it indicates that the graph of the function is concave down at that point. This means that the tangent line, which represents the linear approximation, will lie below the actual graph of the function, leading to an underestimate of f(x).

In contrast, options 2, 3, and 4 do not guarantee an underestimate. Option 2, f′(x)<0, only provides information about the slope of the function, not its curvature. Options 3 and 4 refer to positive values of the first derivative, which indicate increasing slopes, but do not determine the concavity of the graph.