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 `(a)f^(prime)(c)=0` (b) `f^(prime)(c)>0` (c)`f^(prime)(c)<0` (d) none of these

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 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 a) cf^(prime)(c)-f(c)=0 b) cf^(prime)(c)+cf(c)=0 c) f^(prime)(c)-cf(c)=0 d) cf^(prime)(c)+f(c)=0

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

If f(x)a n dg(x) are continuous functions in [a , b] and are differentiable in (a , b) then prove that there exists at least one c in (a , b) for which. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|, w h e r e a

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [a,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 continuous 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 Lagranges mean value theorem.

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 any 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.

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x),g^(prime)(x) be a,b respectively, (a < b) . Show that there exists at least one root of equation f^(prime)(x)g^(prime)(x)+f(x)g(x)=0 on (a , b) .

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