You are given the graph of function f. Determine the relative maxima and relative minima, if any: (If an answer does not exist, enter DNE:) relative minimum smaller x-value (x, Y) relative minimum larger x-value (x, Y) relative maximum (x,Y) =
The Correct Answer and Explanation is:
Correct Answers:
- Relative minimum smaller x-value (x,y)=(−4,0)(x, y) = (-4, 0)(x,y)=(−4,0)
- Relative minimum larger x-value (x,y)=(4,0)(x, y) = (4, 0)(x,y)=(4,0)
- Relative maximum (x,y)=(0,3)(x, y) = (0, 3)(x,y)=(0,3)
Explanation
To identify relative extrema (minimums and maximums) from the graph of a function, we need to observe the peaks and valleys of the graph within the given interval. A relative maximum is a point where the function value is higher than the values immediately around it. A relative minimum is a point where the function value is lower than the values around it.
Looking at the graph provided:
- At x=−4x = -4x=−4:
The function reaches a low point at (x,y)=(−4,0)(x, y) = (-4, 0)(x,y)=(−4,0). As we move away from x=−4x = -4x=−4 toward x=0x = 0x=0, the y-values increase. This indicates that (−4,0)(-4, 0)(−4,0) is a relative minimum because it is lower than its surrounding values. - At x=0x = 0x=0:
The function peaks at (0,3)(0, 3)(0,3). As you move both to the left and to the right from this point, the function values decrease, forming a “peak” at x=0x = 0x=0. Therefore, (0,3)(0, 3)(0,3) is a relative maximum. - At x=4x = 4x=4:
Just like at x=−4x = -4x=−4, the point (4,0)(4, 0)(4,0) is a valley. The function decreases toward this point and then increases afterward. Hence, (4,0)(4, 0)(4,0) is another relative minimum.
In conclusion, the function has two relative minima at (−4,0)(-4, 0)(−4,0) and (4,0)(4, 0)(4,0), and one relative maximum at (0,3)(0, 3)(0,3). These points represent local valleys and a peak within the domain of the function shown.
