Derive the relation
`1/f=1/f_1+1/f_2` for a combination of two thin lenses.

If f (x) satisfies the relation : 2 f(x) +f (1 -x) = x^2 for all real x, then f (x) is :

If f:[1,oo)rarrR:f(x) is monotonic and differentiable function and f(1)=1, then number of solutions of the equation f(f(x))=(1)/(x^(2)-2x+2) is/are……

Find the derivatives of the following at any point of their domains : f(x) =2/(x+1)-x^2/(3x-1) .

Two thin lenses of focal length f_1 and f_2 are in contact and coaxial.The power of the combination is

Two thin lenses of focal length f_1 and f_2 are placed in contant .The focal length of the composite lens will be

If f(x) = {{:([cos pi x]",",x le 1),(2{x}-1",",x gt 1):} , where [.] and {.} denotes greatest integer and fractional part of x, then a. f'(1^(-)) = 2 b. f'(1^(+)) = 2 c. f'(1^(-)) = -2 d. f'(1^(+)) = 0

Give examples of two one-one function f_1 and f_2 from R to R such that f_1 + f_2 : R rarr R defined by : (f_1+f_2)(x) = f_1 (x) + f_2(x) is not one-one.

Show that if f_1 and f_2 are one-one maps from r to R, then the product. f_1 xx f_2: R rarr R defined by (f_1xx f_2) (x) = f_1 (x) xx f_2(x) need not be one-one.

Suppose the function f(x)-f(2x) has the derivative 5 at x=1 and derivative 7 at x=2 . The derivative of the function f(x)-f(4x) at x=1 has the value equal to (a) 19 (b) 9 (c) 17 (d) 14