Question

For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither, b) identity intervals where the function is increasing or decreasing: c) give the coordinates of any points of infection, d) identity intervals where the function is concave up or concave down, and e) sketch the graph. 9x) x²64 a) What are the coordinates of the relative extrema? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice OA. The relative maximum point(s) is/are and there are no relative minimum points) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair Use a comma to separate answers as needed) OB. The relative minimum point(s) is/are and there are no relative maximum point(s) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair Use a comma to separate answers as needed) OC. The relative minimum points) is/are | and the relative maximum point(s)istare (Simplify your answers. Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed.) OD. There are no relative minimum points and there are no relative maximum points b) on what interval(s) is a increasing or decreasing? Select the correct choice below and, if necessary, in in the answer box(es) to complete your choice Er O A The function is increasing on and decreasing on (Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed.) OB. The function is decreasing on The function is never increasing (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed) OC. The function is increasing on The function is never decreasing (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed) OD The function is never increasing or decreasing c) What are the coordinates of the inflection points? Select the correct choice below and, if necessary, fill in the answer box to complete your choice OA. The coordinates of the inflection points are (Type an ordered pair Type an exact answer, using radicals as needed. Use a comma to separate answers as needed) OB. There are no infection points Click to select your answer

Answer 1

Ans, Given function is 16 21x) = x² + 64 (x²+64) x1 - xx 2x (x2+64)2 64 - x2 (8 - x) (8ta (1²+ 64) 2 (x²+6492 q'00 Pu- q() = 0 x = -8 =) = 8 Now (8-) 18+) (²+64)2 q" (a) = X?+64)4 x (-2x) -(64-14) x 2(22+64) x2x (x² + 64) + q" (= -2X (12+64) [ ^2+64 + 128 - 2x2] (x² +64)" (-x² +192) (x²+64)3 - 2n an(x² – 192) (2? +64)3 at x=8 q" (8) 16 (- 128) (128)3 16) is So at x=8 maximum. At x = -2 q" (X) = -168-128 (128)3 So at x = -8 16x) minimum option (c) care -8 Relative Relative and minimum point maximum point He are 8

function f it is increasing decreasing for which q (x) > 0 18 which q (allo Since q (0) q' (1) 70 8 - x)(8 + x) (x² +64) 2 which XE (-8, 8) which * C (-45, - 3) U ( 8 ) for for & q (0) (an option, and The function decreasing on increasin on (-8, 3) (-4,-3) U (8,5) (a (b) option There core no inflection points . (di Since q" (0) = an (n² - 192) (x²+64) 3 2x ( 2-83) (2+2 (3) (x²+64)3 -ve tre -ve tve - 863 0 8.13 So q" () > 0 re (-813,0) U (83") (-0, -83) vlo, 8B) q" (MLDO option (B The function Concave up on and Con Care down om (-88,0) 0 (83) ( - 6 , -8-3) U (9, 33)
The function is concave upword for which double derivative of function is greater than zero.and it is concave downword for which duble derivative is less than zero.