Let f(x) be a polynomial function such that f(3)=3, f'(3)=0 and f"(3)=-3. What is the point (3,3) on the graph y=f(x)?
A. Relative maximum
B. Relative minimum
C. Intercept
D. Inflection point
E. None of these
I got C
To determine the type of point at the coordinates (3,3) on the graph of y=f(x), we need to consider the given information about the function f(x).
Since the point (3,3) lies on the graph of y=f(x), it means that when x = 3, y = 3. This information tells us that the function f(x) passes through the point (3,3).
We are also given the values of the first and second derivatives of f(x) at x = 3:
f'(3) = 0
f"(3) = -3
The first derivative f'(x) of a function represents its rate of change or slope at any given point x. The second derivative f"(x) represents the rate of change or slope of the first derivative. These derivatives help determine the behavior of the function at a specific point.
To classify the point (3,3), we can use the following guidelines:
1. Relative maximum: A point (3, 3) is considered a relative maximum if there is a neighborhood around this point where all other values lie below it. In other words, if f(x) is increasing as x approaches 3 from both the left and right sides, and the function values decrease as x moves away from 3 in either direction, then it would be a relative maximum.
2. Relative minimum: A point (3, 3) is considered a relative minimum if there is a neighborhood around this point where all other values lie above it. In other words, if f(x) is decreasing as x approaches 3 from both the left and right sides, and the function values increase as x moves away from 3 in either direction, then it would be a relative minimum.
3. Intercept: An intercept occurs when the graph of the function intersects the x-axis or y-axis. Since the point (3,3) does not lie on either axis, it cannot be an intercept.
4. Inflection point: An inflection point is a point where the curve changes concavity. If f"(x) changes sign as x approaches 3 from both sides, then it is an inflection point.
By checking the given values of f'(3) and f"(3), we find that f'(3) = 0 and f"(3) = -3, which indicates that the function f(x) has a horizontal tangent line (f'(3) = 0) and is concave down (f"(3) < 0) at x = 3. Therefore, the point (3,3) is an inflection point.
Hence, the correct answer is D. Inflection point.
clearly (3,3) is no kind of intercept!!
Since f'=0 and f"<0, it is a relative max.