Let f(x) be a polynomial function such that f(-2)=5, f'(-2)=0, and f"(-2)=3. The point (-2,5) is a(n)____of the graph f.
A. Relative maximum
B. Relative minimum
C. Intercept
D. Point of Inflection
E. None of these
Relative minumum
rel minimum. f' is zero, and the curve is going up (f")
To determine the nature of the point (-2,5) on the graph of the function f(x), we need to consider the information provided about the derivatives of the function.
Given that f'(-2) = 0 and f"(-2) = 3:
- f'(-2) = 0 is the condition for a stationary point. This means that the graph of the function intersects the x-axis with a horizontal tangent line at x = -2.
- f"(-2) = 3 indicates that the second derivative is positive, which means the graph is concave up.
Based on this information, we can conclude that the point (-2,5) is a relative minimum of the graph of f(x). Therefore, the correct answer is B. Relative minimum.
To determine the nature of the point (-2,5) on the graph of the polynomial function f(x), we need to analyze the given information about its first and second derivatives at that point.
Let's start by understanding what each option represents:
A. Relative maximum: A point where the graph reaches a high point compared to its nearby points.
B. Relative minimum: A point where the graph reaches a low point compared to its nearby points.
C. Intercept: A point where the graph intersects the x-axis or y-axis.
D. Point of Inflection: A point where the concavity of the graph changes (from concave up to concave down or vice versa).
From the information provided, we know the following:
f(-2) = 5: This tells us the y-coordinate of the point (-2,5) on the graph.
f'(-2) = 0: This informs us that the slope of the tangent line to the graph at x = -2 is 0.
f"(-2) = 3: This indicates that the graph is concave upward at x = -2.
Based on this information, we can conclude that the point (-2,5) is a relative minimum of the graph f. This is because the slope of the tangent line is 0 at that point, and the graph is concave upward, indicating that the point represents a low point compared to its nearby points.
Therefore, the answer is B. Relative minimum.