The given function \(f(x) = x^5 - 5x^4 + 5x^3 - 1\) has/have (A) Local maxima at \(x = 1\) (B) Local maximum value is 0 (C) Local minimum at \(x = 3\) (D) Local minimum value is -28 (E) The point of inflexion is \(x = 1\) Choose the correct answer from the options given below

Consider the function \(f(x) = x^{\frac{1}{x}}\). Its

Match List-I with List-II List-I | List-II (A) If \(x = t^2\) and \(y = t^3\) then \(\frac{d^2y}{dx^2}\) at \(t = 1\) | (i) -2 (B) If \(f(x) = \sqrt{x} + 1\), the \(f^{\prime}(1)\) | (ii) -1 (C) The minimum value of \(f(x) = 9x^2 + 12x + 2\) is | (iii) \(\frac{3}{4}\) (D) The poi…

If \(2\begin{bmatrix}a & d\\ b & c\end{bmatrix} + 3\begin{bmatrix}1 & -1\\ 0 & 2\end{bmatrix} = 3\begin{bmatrix}3 & 5\\ 4 & 6\end{bmatrix}\), then the value of \(|a + b - c - d|\) is

The area enclosed by the curve \(y^2 = 4ax\) and its latus rectum is

The system of equations \(3x + 4y = 5, 6x + 7y = -8\) is written in matrix form as

\(\int \frac{xe^x}{(x+1)^2}\,dx =\)

Assume \(P, Q, R\) and \(S\) are matrices of order \(2 \times m\), \(k \times n\), \(m \times 2\) and \(2 \times 3\) respectively. The restrictions on \(k, m\) and \(n\), so that \(PQ + RS\) is defined are

The solution of the differential equation \((x + 1)\frac{dy}{dx} = 1 + y\) is