# 2. The theorem Existence of Extrema on a Closed Interval guarantees thata continuous function on a closed interval takes on both a minimum and amaximum value on the interval. In such conditions those extrema values occurat critical points or endpoints of the interval.Given the function12 x - 2, if - 4x < 65+6x- 2,f(x)if 6 x 8TCa. Graph f(x) and explain why it does not fulfill the assumptions of thetheorem Existence of Extrema on a Closed Interval on -4, 8Will the extrema still exist?b. Find the critical points of f(x), then compare the values of f(x) at thecritical points and the endpoints of -4, 8. What are the largest and smallestof these values?Is the smallest value found in b. the absolute minimum of f(x)? (Hint:compare with f(5.5))d. Does f(x) have both extrema on -4, 8]? Why or why not?No it der nat

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Problem 2 part A, B, and C help_outlineImage Transcriptionclose2. The theorem Existence of Extrema on a Closed Interval guarantees that a continuous function on a closed interval takes on both a minimum and a maximum value on the interval. In such conditions those extrema values occur at critical points or endpoints of the interval. Given the function 12 x - 2, if - 4x < 6 5+6x- 2, f(x) if 6 x 8 TC a. Graph f(x) and explain why it does not fulfill the assumptions of the theorem Existence of Extrema on a Closed Interval on -4, 8 Will the extrema still exist? b. Find the critical points of f(x), then compare the values of f(x) at the critical points and the endpoints of -4, 8. What are the largest and smallest of these values? Is the smallest value found in b. the absolute minimum of f(x)? (Hint: compare with f(5.5)) d. Does f(x) have both extrema on -4, 8]? Why or why not? No it der nat fullscreen
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Step 1

As per norms, three aspects of the question are answered. It is adivisable post multiplie parts separately.  The problem is concerned with continuity, local and global  critical (extreme values) ,

Step 2

part 1) graph of f(x) over the closed interval [-4,8]

Step 3

Part 2 )f(x) fails the continuity test at x=6, f(x) approaches -18 from the left and 5 (=f(6)) from the right...

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### Calculus 