It is given that,
\begin{align} \frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}\end{align}
∴ Anti derivative of
\begin{align} 4x^3 - \frac{3}{x^4} = f(x)\end{align}
∴ \begin{align} f(x)= \int \left(4x^3 - \frac{3}{x^4}\right).dx\end{align}
\begin{align} f(x)= 4\int x^3.dx - 3\int {x^{-4}}.dx\end{align}
\begin{align} f(x)= 4\left(\frac {x^4}{4}\right) - 3\left(\frac {x^{-3}}{-3}\right) + C\end{align}
∴ \begin{align} f(x)= x^4 + \frac{1}{x^3} + C\end{align}
Also, f(2) = 0
∴ \begin{align} f(2) =\left(2\right)^4 + \frac{1}{\left(2\right)^3} + C = 0 \end{align}
=> \begin{align} 16 + \frac{1}{8} + C = 0 \end{align}
=> \begin{align} C = -\left(16 + \frac{1}{8}\right) \end{align}
=> \begin{align} C = \frac{-129}{8} \end{align}
∴ \begin{align} f(x)= x^4 + \frac{1}{x^3} -\frac{129}{8} \end{align}
Hence, the correct answer is A.
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.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f –1 and show that (f –1)–1 = f.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by
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.