y = x2 + 2x + C : y' - 2x - 2 = 0
y = x2 + 2x + C
Differentiating both sides of this equation with respect to x, we get:
\begin{align}y^{'}=\frac{d}{dx}(x^2 + 2x + C)\end{align}
=> y' = 2x + 2
Substituting the value of y' in the given differential equation, we get:
L.H.S. = y' - 2x - 2 = 2x + 2 - 2x - 2 = 0 = R.H.S.
Hence, the given function is the solution of the corresponding differential equation.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ?
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
y = cosx + C : y' + sinx = 0
The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?