If the function f(x) = 2x³ - 9ax² + 12a + 1, where a > 0, attains its maximum and minimum values at p and q respectively such that p² = q, then a equals

• 3

• 1

• 2

• $\frac{1}{2}$

For the function 3x⁴ - 8x³ + 12x² - 48x + 25 in the interval [1, 3]. The value of maxima and minima are

• 16, - 39

• - 16, 39

• 6, - 9

• None of these

f(x) = x³ - 27x + 5 is an increasing function when

• x < - 3

• $\left|\mathrm{x}\right| >\hspace{0.17em}3$

• $\mathrm{x} \le - 3$

• $\left|\mathrm{x}\right| <\hspace{0.17em}3$

In the interval $\left(0, \frac{\mathrm{\pi }}{2}\right)$ function log(sin(x)) is

• increasing

• decreasing

• neither increasing nor decreasing

• None of the above

A cone of maximum volume is being cut from sphere, then ratio between height of cone and diameter of sphere is

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{3}{4}$

• $\frac{1}{4}$

The function x⁵ - 5x⁴ + 5x³ - 10 has a maximum, when x is equal to

• 3

• 2

• 1

• 0

The function y = 2x³ - 9x² + 12x - 6 is monotonic decreasing when

• 1 < x < 2

• x > 2

• x < 1

• None of these

Maximum slope of the curve y = - x³ + 3x² + 9x - 27 is

• 0

• 12

• 16

• 32

The points on the curve x² = 2y which are closest to the point (0, 5) are

• (2, 2), (- 2, 2)

• $\left(2\sqrt{2}, 4\right), \left(- 2\sqrt{2}, 4\right)$

• $\left(\sqrt{6}, 3\right), \left(- 6\sqrt{3}, 3\right)$

• $\left(2\sqrt{3}, 6\right), \left(- 2\sqrt{3}, 6\right)$

The point on the curve y = 2x² - 4x + 5, at which the tangent is parallel to x-axis, will be

• (1, 3)

• (- 1, 3)

• (1, - 3)

• (- 1, - 3)