Show that the equation has exactly one real solution.8x + cos(x)=0We first show that the given equation has at least one real solution.The sentences in the following scrambled list can be used to prove the statement when they are put in the correct order.By the Intermediate Value Theorem, there is a number c in the interval (,0) such that f(c)=0.Since f is the sum of the polynomial 8x and the trigonometric function cos(x), f is continuous and differentiable for all x.Thus, the given equation has at least one real solution.We construct a proof by selecting appropriate sentences from the list and putting them in the correct order.Let f(x)=8x + cos(x),Then f()=81<0 and f(0)=1>0.---Select--- By the Intermediate Value Theorem, there is a number c in the interval (,0) such that f(c)=0. Since f is the sum of the polynomial 8x and the trigonometric function cos(x), f is continuous and differentiable for all x. Thus, the given equation has at least one real solution.---Select--- By the Intermediate Value Theorem, there is a number c in the interval (,0) such that f(c)=0. Since f is the sum of the polynomial 8x and the trigonometric function cos(x), f is continuous and differentiable for all x. Thus, the given equation has at least one real solution.---Select--- By the Intermediate Value Theorem, there is a number c in the interval (,0) such that f(c)=0. Since f is the sum of the polynomial 8x and the trigonometric function cos(x), f is continuous and differentiable for all x. Thus, the given equation has at least one real solution.We continue the proof by showing that the given equation has exactly one real solution.The sentences in the following scrambled list can be used to prove the statement when they are put in the correct order.Since f is continuous on [a, b] and differentiable on (a, b), Rolle's theorem implies that there is a number r in (a, b) such that f(r)=0.This contradiction shows that the given equation can't have two distinct real solution, so it has exactly one solution.But f(r)=8 sin(r)>0 since sin(r)1.We construct a proof by selecting appropriate sentences from the list and putting them in the correct order.Suppose the equation has distinct real solutions a and b with a < b, the f(a)= f(b)=0.---Select--- Since f is continuous on [a, b] and differentiable on (a, b), Rolle's theorem implies that there is a number r in (a, b) such that f(r)=0. This contradiction shows that the given equation can't have two distinct real solution, so it has exactly one solution. But f(r)=8 sin(r)>0 since sin(r)1.---Select--- Since f is continuous on [a, b] and differentiable on (a, b), Rolle's theorem implies that there is a number r in (a, b) such that f(r)=0. This contradiction shows that the given equation can't have two distinct real solution, so it has exactly one solution. But f(r)=8 sin(r)>0 since sin(r)1.---Select--- Since f is continuous on [a, b] and differentiable on (a, b), Rolle's theorem implies that there is a number r in (a, b) such that f(r)=0. This contradiction shows that the given equation can't have two distinct real solution, so it has exactly one solution. But f(r)=8 sin(r)>0 since sin(r)1.