suppose f(x) is a function such that f(3)=5, f'(3)=0, and f''(3)=3. Is the point (3, 5) a relative maximum, minimum or inflection point?

f ' (3) = 0 tells me that at the point (3,5) the tangent is horizontal, so (3,5) must be either a maximum or a minimum.

f '' (3) = 3 tell sme that at the point (3,5) the curve is concave up, so

(3,5) must be a minimum point.

(it couldn't possibly be a point of inflection or else the f '' (3) would have been 0 )

To determine whether the point (3, 5) is a relative maximum, minimum, or inflection point, we can analyze the behavior of the function using derivatives.

1. Start by analyzing the first derivative, f'(x):
- Since f'(3) = 0, it indicates that the slope of the curve at x = 3 is horizontal (i.e., the tangent line is parallel to the x-axis).
- This suggests a possible turning point or extremum near x = 3.

2. Consider the second derivative, f''(x):
- Since f''(3) = 3, it indicates that the concavity of the curve is positive (i.e., the curve is concave up) near x = 3.
- Positive concavity suggests that the curve might have a relative minimum.

Based on this information, the point (3, 5) is a relative minimum. Here's the reasoning behind it:

- When f'(x) = 0 and f''(x) > 0 at a point, it indicates there's a local minimum.
- Since f(3) = 5 and the derivative values support a minimum at x = 3, the point (3, 5) qualifies as a relative minimum.

Note that we have made assumptions based on the given information about the derivatives at x = 3. To fully confirm the nature of the point, plotting the function or analyzing further points around x = 3 may be necessary.