Statement 1: If `f(x)` is an odd function, then `f^(prime)(x)` is an even function.

Statement - I : If f(x) is an odd function, then f'(x) is an even function. Statement - II : If f'(x) is an even function, then f(x) is an odd function.

If f(z) is an odd function, prove that, int_(a)^(x)f(z)dz is an even function.

If f(t) is an odd function, then int_(0)^(x)f(t) dt is -

If f(x) is an odd function then int_(-a)^(a)f(x) is equal to-

Let G(x)=(1/(a^x-1)+1/2)F(x), where a is a positive real number not equal to 1 and f(x) is an odd function. Which of the following statements is true? (a) G(x) is an odd function (b) G(x)i s an even function (c) G(x) is neither even nor odd function. (d)Whether G(x) is an odd or even function depends on the value of a

Let f(x) be a differentiable even function, consider the following statements : (i) f'(x) is an even function. (ii) f'(x) is an odd function. (iii) f'(x) may be even or odd. Which of the above statements is/are correct ?

STATEMENT 1: int_a^xf(t)dt is an even function if f(x) is an odd function. STATEMENT 2: int_a^xf(t)dt is an odd function if f(x) is an even function.

Let f be a differential function such that f(x)=f(2-x) and g(x)=f(1 +x) then (1) g(x) is an odd function (2) g(x) is an even function (3) graph of f(x) is symmetrical about the line x= 1 (4) f'(1)=0

If f(t) is an odd function, then prove that varphi(x)=int_a^xf(t)dt is an even function.

If f(x) satisfies the relation f(x+y)=f(x)+f(y) for all x , y in R and f(1)=5,t h e n (a) f(x) is an odd function (b) f(x) is an even function (c) sum_(r=1)^mf(r)=5^(m+1)C_2 (d) sum_(r=1)^mf(r)=(5m(m+2))/3