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

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)

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)

The least value of 6tan^2varphi+54cot^2varphi+18 is 54 when A.M. geq GM. Is applicable for 6tan^2varphi,54cot^2varphi,18 54 when A.M. geq GM. Is applicable for 6tan^2varphi,54cot^2varphi,18 is added further 78 when tan^2varphi=cot^2varphi (I) is correct, (II) is false (I) and (II) are correct (III) is correct None of the above are correct

If f(x)=|x-a|varphi(x) , where \ varphi(x) is continuous function, then f'(a^+)=varphi(a) (b) f^(prime)(a^-)=-varphi(a) (c) f^(prime)(a^+)=f'(a^-) (d) none of these

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