If D =diag. `[d_(1), d_(2),..., d_(n)]`, then prove that f(D) = diag. `[f(d_(1)), f(d_(2)), ... , f(d_(n))]`, where f(x) is a polynominal with scalar coefficieents.

If D=d i ag[d_1, d_2, d_n] , then prove that f(D)=d i ag[f(d_1),f(d_2), ,f(d_n)],w h e r ef(x) is a polynomial with scalar coefficient.

Statement 1: if D= diag [d_1, d_2, ,d_n] ,then D^(-1)= diag [d_1^(-1),d_2^(-1),...,d_n^(-1)] Statement 2: if D= diag [d_1, d_2, ,d_n] ,then D^n= diag [d_1^n,d_2^n,...,d_n^n]

Using the first principle, prove that: d/(dx)(f(x)g(x))=f(x)d/(dx)(g(x))+g(x)d/(dx)(f(x))

If f(x+1)+f(x-1)=2f(x)a n df(0),=0, then f(n),n in N , is nf(1) (b) {f(1)}^n (c) 0 (d) none of these

if f(x) =ax +b and g(x) = cx +d , then f[g(x) ] - g[f(x) ] is equivalent to

Using the first principle, prove that d/(dx)(1/(f(x)))=(-f^(prime)(x))/([f(x)]^2)

Statement 1: If differentiable function f(x) satisfies the relation f(x)+f(x-2)=0AAx in R , and if ((d/(dx)f(x)))_(x=a)=b ,t h e n((d/(dx)f(x)))_(a+4000)=bdot Statement 2: f(x) is a periodic function with period 4.

If (d)/(dx)[f(x)]=sinx+cosx , (i) Find f(x) (ii) Write f(x) , when f(x)=1

If f(x)=cos^2theta+sec^2theta, then f(x) f(x)>1 (d) f(x)geq2

If f (x) =(1)/(x +1) then its differentiate is given by :