The ordinate of the point on the curve $y^2 = 8x$ for which abscissa and ordinate change at the same rate is:

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

The function $f(x) = \frac{4x^2+1}{x}$ is decreasing in

(A). $\left(\frac{-1}{2}, \frac{1}{2}\right)$

(B). $\left(-\infty, \frac{-1}{2}\right)$

(C). $\left(0, \frac{1}{2}\right)$

(D). $\left(\frac{-1}{2}, 0\right)$

Choose the correct answer from the options given below:

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

The slope of the normal to the curve $y = 2x^2 - 4$ at P (1, -2) is :

The function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$

(A) is increasing on $\left[0, \frac{\pi}{6}\right]$

(B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$

(C) is increasing on $\left[0, \frac{\pi}{2}\right]$

(D) is decreasing on $\left[0, \frac{\pi}{2}\right]$

Choose the correct answer from the options given below:

If the function $f(x)$ defined by $f(x) = x^3 - 3x + 100, (x \in \mathbb{R})$, (Where $\mathbb{R}$ is set of real numbers), then $f(x)$ has