Question

let f be a differentiable function with f(1)=−2. the graph of f′, the derivative of f, is shown above. which of the following statements is true about the line tangent to the graph of f at x=1 ?

Answer 1

The statement about the tangent line at (x=1) depends on the behavior of (f') at that point. If (f'(1)) is positive, the tangent line has a positive slope; if (f'(1)) is negative, the slope is negative. If (f'(1) = 0), the tangent line is horizontal.

Explanation:

The behavior of the tangent line to the graph of (f) at (x=1) is determined by the sign of the derivative (f') at that point. If (f'(1)) is positive, it indicates that the function (f) is increasing at (x=1). In this case, the tangent line to the graph of (f) at (x=1) will have a positive slope, reflecting the upward direction of the function at that specific point.

Conversely, if (f'(1)) is negative, it implies that (f) is decreasing at (x=1), leading to a tangent line with a negative slope, representing the downward trend of the function. If (f'(1)) equals zero, the tangent line will be horizontal, signifying a point of inflection or a local extremum in the graph of (f) at (x=1).

This occurs when the function transitions from increasing to decreasing or vice versa. Therefore, the behavior of the tangent line at (x=1) directly mirrors the nature of the function at that specific point, providing insights into the local characteristics of the graph.