A particle moves along a curve $3y^2 = 6x^3 + 8$ such that its y-coordinate is changing at the rate of 9 cm/sec at point (1,2). Then the rate of a change of x-coordinate at the same point is :

The tangent to the curve $y^2 + 2x - 5 = 0$ at the point (h, k) is parallel to the line $x + 2y = 4$, then the value of 'k' is:

For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(C) $f(x)=5x^4-3x^2+e$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is

The function $f(x) = x^4 - 2x^2$ is increasing on

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE?

(A) $f'(x) = 3(x - 1)(x - 3)$

(B) The critical points of the function are $x = 1$ and $x = 3$

(C) $x = 1$ is the point of local minimum

(D) The local maximum value is $-4$

Choose the correct answer from the options given below:

The interval in which the function $f(x)=x^4-\dfrac{x^3}{3}$ is increasing, is