Question

Show that the function flx)- x+8x+5 has exactly one zero in the interval [-1, 01. Which theorem can be used to determine whether a function f(x) has any zeros a given interval? O A. Extreme value theorem O B. Intermediate value theorem OC. Rolle's Theorem O D. Mean value theorem apply this theorem, evaluate the function fix)x +8x+5 teach endpoint of the interval [-1, 01 f-1)(Simplify your answer.) f(0) (Simplify your answer.) According to the intermediate value theorem, f(x) x +8x + 5 has in the given interval. Now, determine whether there can be more than one zero in the given interval. Rolle's Theorem states that for a function f(x) that is continuous every point over the closed interval [a,bl and differentiable at every point of its interior (a,b), if f(a) f(b), then there is at least one number c in (a,b) at which f(e) 0 Find the derivative of fix)-x +8x + 5. f fix) be zero in the interval [- 1, O1? Can the derivative No Yes The function fx) x+8x+5 has at least one zero at some point x a in the interval -1, 0. According to Rolle's Theorem, can there e another point x b in this interval where f(a) f(b) - 0? O Yes O No Thus, since the intermediate value theorem shows that f(x)-x"+8x +5 has at least one zero in the interval [-1, 0] and Rolle's Theorem shows that there cannot be two points x exactly one zero in the inteerval [-1,01. a and x b for which f(a) f(b) in this interval, the function f(x) has

Answer 1

Grivam LE Fatr- f)= (8)+5 8+5=2 -5 -oO tle Honceheu isoue rootBl x--1 to x-0 Juorm Intermedhabs Val Hueoren (-) ()45- fl)= + = Oto 5- Acc. to Tntermediot valu um has intweol gium at ecst ouroot Riivabtue f

de 4x2+80 259 # ND Auy Rollag Hm f) = a+8a +5 b8b+5 O-4 fl0)=5 Ture conl AgiNo Le ano ter- b