Question

3.pdf?X-Amz-Algorithm 7. Consider the function f(x) = + List all intervals on which f is increasing, and list all intervals in which f is decreasing. (You must show the right calculus steps to get full credit. Just taking the answers from a graph is not enough.) 8. Consider the function g(x) = x In x on the interval [0.1, e) For each part of this question, fill in the blank. Use exact values (no rounding) A. The critical point(s) of g are located at x - B. Use the First Derivative Test: There is a local atx C. The absolute minimum value of g on this interval is D. The absolute maximum value of g on this interval is

Answer 1

2 15 + 20 15 5 for increasing a >o orc = .5708 ② f(x) = x - 8x² + 8x 20 5 Answer f (x) = 4 x ² 8x34² 8x2a 8x² + 164 5 , f'(x) = ** - 3x + 15% is x (x2 - 8x +16) yo x (x-4)²70 x 70 lie., f is increasing on interval a € (0,0) for decreasing a f'(x) = x - 8x² + 16x co i.e.e x (x-4) 2 co f is decreasing on interval x 6 (-0,0) [ool, e] A. g'() = x. 1/4+ 1.lox= It enx = 0 = ena=-| which gives the solution, this g(n) has a critual pointo at 5 5 = xco 8 ) g(x) = x lng > e-l=h x=e-1 x=/e-1 6 = 94(1) = d. an (It lns) = 0+1 L x 92 6. the 니 Inlo 10 At x=61 9 9" (e-1) = 1 / 4 = e yo (min) ie, there is a local (minimum at x = le- g(0:1) = 0.1 ln (0.1) = --O. Bos g(e) = C line = e = 2.118 (absolute max.) g (e-) = e 'ene!--et=-0.3674 Cabsolute mine) the abs. min. value of g on this interval is NON-é The abs. max. Value of on this internal is banore g