The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

A steel plant is capable of producing 'x' tonnes per day of low-grade steel and 'y' tonnes per day of high-grade steel, where $y=\frac{32-5x}{10-x}$. If the fixed price of low-grade steel is half that of high-grade steel, then the quantity of low-grade steel that should be produced per day for maximum receipts is

The interval in which the function $f(x) = \log_e(2+x) - \frac{2x}{2+x}$ is increasing is

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis?

(A) The point of contact is (11, 0).

(B) The equation of tangent is $x - 72y - 11 = 0$

(C) The equation of normal is $72x + y - 11 = 0$

(D) The slope of the tangent at the given point of contact is $\frac{1}{88}$

Choose the correct answer from the options given below:

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

The function $f(x) = \sin x + \cos x$; $0 \leq x \leq 2\pi$ is

(A) Increasing in $\left(0, \frac{\pi}{4}\right)$

(B) Decreasing in $(0, \pi)$

(C) Increasing in $\left(\frac{5\pi}{4}, 2\pi\right)$

(D) Decreasing in $\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$

(E) Neither increasing Nor decreasing in $\left(0, \frac{\pi}{4}\right)$

Choose the correct answer from the options given below:

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is