Question

the function $f(x)$ is graphed below. how many points on the graph represent a relative minimum?
graph of a function with points a, b, c: a is a local maximum on the left, b is a local minimum, c is an x-intercept on the right.
answer
attempt 1 out of 2

Explanation:

Step1: Define relative minimum

A relative minimum of a function is a point where the function changes from decreasing to increasing (the graph "bottoms out" at that point, and the function value at that point is less than or equal to the values of the function in its immediate neighborhood).

Step2: Analyze each point

• Point \( a \): The graph has a peak at \( a \) (it changes from increasing to decreasing), so this is a relative maximum, not a minimum.
• Point \( b \): The graph changes from decreasing to increasing at \( b \) (the graph comes down to \( b \) and then goes up), so this is a relative minimum.
• Point \( c \): This is an \( x \)-intercept (where the graph crosses the \( x \)-axis), and the graph is increasing through \( c \), so it is not a relative minimum (the function values around \( c \) are not both higher than at \( c \); in fact, to the left of \( c \) the function is increasing towards \( c \) from \( b \), and to the right it continues increasing, so \( c \) is not a minimum point in its neighborhood).

Answer:

1