Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. x² - 10x + 65 X-8 Step 1 Begin by finding the intercepts. (If an answer does not exist, enter DNE.) y = To find the y-intercept, substitute x = 0 and solve for y. The y-intercept is 65 ). 8 To find the x-intercept, substitute y = 0 and solve for x. Step 2 The x-intercept is (x, y) = (0, (x, y) = (DNE y' = Next, differentiate the given function y = (2x 10) (x - 8) (x - 8) (x - 1)(x - 15) 2 (x - 8) y" = 65 8 Step 5 The derivative y' = 0 when x = 15,1 relative maximum 0, relative minimum DNE 98 inflection point Step 3 To determine maximums and minimums by the Second Derivative Test, we differentiate y' with respect to x. 98 (x - 8)³ Substituting x = 15 into y", we see that y"> Substituting x = 1 into y", we see that y" x²2 10x + 65 2 2 (x-8) There is no value of x for which y" is zero. x² - 10x + 65 and write as a single fraction completely factored. x - 8 Step 4 Therefore by the Second Derivative Test, give the following points. (If an answer does not exist, enter DNE.) (x, y) = (1,- 8 (x, y) = (15,20 (x, y) = (DNE Submit Skip (you cannot come back) 1, 15 0. (entered as a comma-separated list). 0. 1.-8 15, 20 DNE Next, we look at the asymptotes. To find the horizontal asymptote, consider the limit of the function y = x² − 10x + 65 when x→ +∞. Since the power of the numerator is greater than the power of the denominator, this limit does not exist X-8 To find the vertical asymptotes, we find values of x for which the function is not defined. Note that the denominator equals 0 when x = 8 To find the slant asymptote, we can use long division to divide the numerator x² - 10x + 65 by the denominator x - 8 to simplify the improper rational expression. The quotient is and the remainder is 49.