Calculating f, f', and f'' from a Graph at Specific Points

Math Problem Statement

Here is a graph of a function y = f(x) with three points labeled A, B, and C. For point A, decide whether f, f', and f'' are positive, negative, or 0. Give your answer in parentheses as a triple of symbols p, n, and 0 (for 'positive', 'negative', and '0'). For example, an answer (n, p, 0) indicates that f < 0, f' > 0, and f'' = 0 at the point.

Solution

It seems you're working with a graph of a function and need to identify the values of $f$ , $f'$ , and $f''$ at point C.

From the second image, I can see the graph and points A, B, and C. To answer this for point C:

$f(C)$ : This is the value of the function at point C. Since the point is at the x-axis (y = 0), $f(C) = 0$ .
$f'(C)$ : This represents the slope of the tangent line to the curve at point C. Since the curve is flat at point C (the slope is 0), $f'(C) = 0$ .
$f''(C)$ : This is the curvature (the concavity) of the function at point C. As point C is at an inflection point (changing direction from concave down to concave up), the second derivative is $f''(C) = 0$ .

So, the answer for point C should be (0, 0, 0).

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Concavity

Formulas

f(x), f'(x), f''(x)

Theorems

First and second derivative tests

Suitable Grade Level

Grades 11-12

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