f is continuous in [a, b] and differentiable in (a, b) (where a>0 ) such that `f(a)/a=f(b)/b`. Prove that there exist `x_0 in (a, b)` such that `f'(x_0 ) = f(x_0)/x_0`

Let f be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

Let f be arry continuously differentiable function on [a,b] and twice differentiable on (a.b) such that f(a)=f(a)=0 and f(b)=0. Then

Let f be continuous on [a, b] and assume the second derivative f" exists on (a, b). Suppose that the graph of f and the line segment joining the point (a,f(a)) and (b,f(b)) intersect at a point (x_0,f(x_0)) where a < x_0 < b. Show that there exists a point c in(a,b) such that f"(c)=0.

If f is continuous on [a;b] and f(a)!=f(b) then for any value c in(f(a);f(b)) there is at least one number x_(0) in (a;b) for which f(x_(0))=c

Let f:[a,b]rarr R be a function,continuous on [a,b] and twice differentiable on (a,b). If,f(a)=f(b) and f'(a)=f'(b), then consider the equation f''(x)-lambda(f'(x))^(2)=0. For any real lambda the equation hasatleast M roots where 3M+1 is

Let f be continuous on [a;b] and differentiable on (a;b) If f(x) is strictly increasing on (a;b) then f'(x)>0 for all x in(a;b) and if f(x) is strictly decreasing on (a;b) then f'(x)<0 for all x in(a;b)

If f''(x) le0 "for all" x in (a,b) then f'(x)=0

Let f be a function from [a,b] to R , (where a, b in R ) f is continuous and differentiable in [a, b] also f(a) = 5, and f'(x) le 0 for all x in [a,b] then for all such functions f, f(b) + f(lambda) lies in the interval where lambda in (a,b)