If `f` is a continuous function on `[0,1],` differentiable in (0, 1) such that `f(1)=0,` then there exists some `c in (0,1)` such that `cf^(prime)(c)-f(c)=0` `cf^(prime)(c)+cf(c)=0` `f^(prime)(c)-cf(c)=0` `cf^(prime)(c)+f(c)=0`

Let fa n dg be differentiable on [0,1] such that f(0)=2,g(0)=0,f(1)=6a n dg(1)=2. Show that there exists c in (0,1) such that f^(prime)(c)=2g^(prime)(c)dot

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

If f(x) is continuous in [a , b] and differentiable in (a,b), then prove that there exists at least one c in (a , b) such that (f^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

If f(x) is continuous in [a , b] and differentiable in (a,b), then prove that there exists at least one c in (a , b) such that (f^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

Let f be a continuous function defined onto [0,1] with range [0,1]. Show that there is some c in [0,1] such that f(c)=1-c

Leg f be continuous on the interval [0,1] to R such that f(0)=f(1)dot Prove that there exists a point c in [(0,1)/2] such that f(c)=f(c+1/2)dot

Let f be a continuous, differentiable, and bijective function. If the tangent to y=f(x)a tx=a is also the normal to y=f(x)a tx=b , then there exists at least one c in (a , b) such that f^(prime)(c)=0 (b) f^(prime)(c)>0 f^(prime)(c)<0 (d) none of these

Statement-1: If a continuous funtion on [0,1] satisfy 0 le f(x) le 1 , then there exist c in [0,1] such that f(c )=c Statement-2: lim_(x to c) f(x)=f(c)

Let f(x)be continuous on [a,b], differentiable in (a,b) and f(x)ne0"for all"x in[a,b]. Then prove that there exists one c in(a,b)"such that"(f'(c))/(f(c))=(1)/(a-c)+(1)/(b-c).