sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f…
sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f. (enter your answers as a comma - separated list. if an answer does not exist, enter dne.)\nf(x)=\begin{cases}x^{2}&\text{if } - 1leq xleq0\\8 - 9x&\text{if }0lt xleq1end{cases}\nabsolute maximum value\nabsolute minimum value\nlocal maximum value(s)\nlocal minimum value(s)
Answer
Explanation:
Step1: Analyze (y = x^{2}) on ([-1,0])
The function (y=x^{2}) is a parabola opening upwards with vertex at ((0,0)). When (x=-1), (y = (-1)^{2}=1); when (x = 0), (y=0).
Step2: Analyze (y=8 - 9x) on ((0,1])
This is a linear - function with a slope (m=-9). When (x = 0^{+}), (y=8) (approaching from the right), and when (x = 1), (y=8-9\times1=-1).
Step3: Check endpoints and critical points
The left - hand endpoint of the domain ([-1,1]) is (x=-1) with (f(-1)=1), the right - hand endpoint is (x = 1) with (f(1)=-1), and the point where the two sub - functions meet at (x = 0), (f(0)=0) from (y = x^{2}) and approaches (8) from (y=8 - 9x) as (x\to0^{+}).
Step4: Determine maximum and minimum values
The function is continuous on ([-1,1]) (from the left at (x = 0) for (y=x^{2}) and from the right at (x = 0) for (y = 8-9x)). The absolute maximum value occurs at (x = 0^{+}) and (f(0^{+})=8). The absolute minimum value occurs at (x = 1) and (f(1)=-1). There are no local maxima or minima other than the absolute ones in this case.
Answer:
absolute maximum value: (8) absolute minimum value: (-1) local maximum value(s): DNE local minimum value(s): DNE