Let f(x) be a polynomial function such that f(-2)=5, f’(-2)=0 and f”(-2)=3. The point (-2, 5) is which of the following for the graph of f?
A. relative maximum
B. relative minimum
C. intercept
D. inflection point
E. none of these
since f'(-2)=0 and f"(-2)>0, (-2,5) is a minimum. (f is concave up there)
To determine the nature of the point (-2, 5) on the graph of the polynomial function f(x), we need to analyze the information given about its derivative and second derivative at x = -2.
Given that f'(-2) = 0, we know that the slope of the tangent line to the graph of f(x) at x = -2 is 0.
If the second derivative, f''(-2), is positive, then the graph changes concavity from concave down to concave up at x = -2, indicating that the point (-2, 5) is an inflection point.
If the second derivative, f''(-2), is negative, then the graph changes concavity from concave up to concave down at x = -2, indicating that the point (-2, 5) is also an inflection point.
However, since f''(-2) = 3, which is positive, the graph changes concavity from concave down to concave up at x = -2. Therefore, the given point (-2, 5) is not an inflection point.
Now, let's analyze whether the point (-2, 5) is a relative maximum or a relative minimum.
Since f(-2) = 5, the y-coordinate of the point (-2, 5) is 5.
If f''(-2) > 0 and the value of f'(x) changes from positive to negative as x approaches -2 from the left, then the point (-2, 5) is a relative maximum.
If f''(-2) > 0 and the value of f'(x) changes from negative to positive as x approaches -2 from the left, then the point (-2, 5) is a relative minimum.
Since f'(-2) = 0, which means that the value of f'(x) does not change sign as x approaches -2, we cannot determine whether the point (-2, 5) is a relative maximum or a relative minimum based on the given information.
Therefore, the correct answer is E. none of these.