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

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

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [1,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuou and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranes mean value theorem.

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'(c))/(3c^(2))=(f(b)-f(a))/(b^(3)-a^(3))

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

Statement 1: If 27 a+9b+3c+d=0, then the equation f(x)=4a x^3+3b x^2+2c x+d=0 has at least one real root lying between (0,3)dot Statement 2: If f(x) is continuous in [a,b], derivable in (a , b) such that f(a)=f(b), then there exists at least one point c in (a , b) such that f^(prime)(c)=0.

If f(x) is continuous in [a, b] and differentiable in (a, b), prove that there is atleast one c in (a, b) , such that (f'(c))/(3c^(2))= (f(b)-f(a))/(b^(3)-a^(3)) .

Let f: R-> be a differentiable function AAx in R . If the tangent drawn to the curve at any point x in (a , b) always lies below the curve, then (a) f^(prime)(x)<0,f^(x)<0AAx in (a , b) (b) f^(prime)(x)>0,f^(x)>0AAx in (a , b) (c) f^(prime)(x)>0,f^(x)>0AAx in (a , b) (d) non eoft h e s e

Let f be a twice differentiable function such that f(a)=f(b)=0 and f(c)>0 for a