If `varphi(x)` is differentiable function `AAx in R` and `a in R` such that `varphi(0)=varphi(2a),varphi(a)=varphi(3a)a n dvarphi(0)!=varphi(a)` then show that there is at least one root of equation `varphi^(prime)(x+a)=varphi^(prime)(x)in(0,2a)`

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

If varphi(x) is a polynomial function and varphi^(prime)(x)>varphi(x)AAxgeq1a n dvarphi(1)=0, then varphi(x)geq0AAxgeq1 varphi(x)

If varphi(x)=1/(1+e^(-x))a n dS=varphi(5)+varphi(4)+varphi(3)++varphi(-3)+varphi(-4)+varphi(-5) , then the value of S is. a. 5 b. 11/2 c. 6 d. 13/2

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.

The general solution of the differential equation, y^(prime)+yvarphi^(prime)(x)-varphi^(prime)(x) varphi(x)=0 , where varphi(x) is a known function, is

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

If y(x) satisfies the differential equation y^(prime)-ytanx=2xs e c x and y(0)=0 , then

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) (d) none of these

Let f(x)a n dg(x) be two differentiable functions in Ra n df(2)=8,g(2)=0,f(4)=10 ,a n dg(4)=8. Then prove that g^(prime)(x)=4f^(prime)(x) for at least one x in (2,4)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