Question

(5 points) A continuous function f, defined for all x, has the following properties: 1. f is decreasing 2. f is concave up 3. f(26) = -5 4. f'(26) = - Sketch a possible graph for f, and use it to answer the following questions about f. A. For each of the following intervals, what is the minimum and maximum number of zeros f could have in the interval? (Note that if there must be exactly N zeros in an interval, the minimum and maximum are both N.) minimum maximum -00 < x < 0 0 < x < 1 1<x< 26 26 <<< B. Are any of the following possible values for f'(1)? (Enter your answer as a comma-separated list, or enter 'none' if none of them are possible.) -3, -2, -1, – 6,0, 6, 1, 2, 3. possible values: f'(1) = c. What happens to f as x + - ? lim f(x) = infinity

Answer 1

the x-axi at STEPI Giren that f is increasing If the function is increasing throughout & domain, it can have a zero or have none of we cut any pt. we cannot cut it another pt. as this function never decreases f is concave down :- If the function is concave down at all pt. of the function, the tangent line is above the function, f126) = -5 -s. The pt. (26,-5) pt. (26.-5) belongs to the graph of the function

at this pt. line f'(26) = -13 Find the tangent Tine y = matn m is the slope of the tangent the first denvative evaluated at the pt. m=-4/5 y = 3x tn. -5= -7/226th } *26+n & evaluate in the pt. .. - = - 33 n= -5 +26 .. n = 26-25 .: y = -x + 1 & tangent line ean 3- 2+ cois) o 3 -2 For past (a) we know that such a curve can mly C003 hre axis one from o o to too We albo know that it

must cross X-axis somewhere after x=1, we can figure out by assuming slope is less than & for allipts less than 26. since a line going through (26, & with a slope of & would coors the X-axi at (lo) min. max 0 -00<xso 0<x al 1<x 326 1 1 0 0 26<«< For pasteb) as we know that before 1, the curve hasn't crossed the x-axi.yet so f'(1) must be negative Beyond that we can't really tell anything about for so we'll just say frent anny negative no. is valid for 1. Note that because a curve and not a line fill can't be o. It has to be less frana fins a (3,-2,-1! welve