From the graph \(y = 4{x^2} - {x^3}\), the co-ordinates of the point of inflexion are?

The adjacent graph shows the sketch of \(y = {x^4} - 2{x^3}\). The curve is concave down in the domain?

From the graph, \(y = {x^4} - 2{x^3}\), the curve increases in the domain?

The adjacent graph shows the sketch of \(y = {x^4} - 4{x^2}\). The curve increases in the domains?

For the function \(y=x^4-2x^3\)

(i)  Find the coordinates of the points where the curve crosses the axes

(ii)  Find the coordinates of the stationary points and determine their nature

(iii)  Find the coordinates of the points of inflection

(iv) Sketch the graph of \(f(x)\) clearly indicating the intercepts, stationary points, and points of inflection

Consider the curve \(y = {x^4} - 8{x^2} + 16\)

(i)  Show that \(\dfrac{{dy}}{{dx}} = 4x(x - 2)(x + 2)\)

(ii)  Find the stationary points on the curve and determine their nature

(iii)  Sketch the curve showing all intercepts on the axes and stationary points

For the function \(p(x) = (x - 2)({x^2} + 1)\)

(i)  find the coordinates of the turning points of \(p(x)\) and state their nature

(ii)  Draw a sketch of \(y = p(x)\) in the domain \(0 \le x \le 3\)

Consider the curve given by \(y = 3{x^4} - 12{x^3} + 12{x^2} + 2\)

(i)  Find the coordinates of the stationary points

(ii)  Find all values of \(x\) for which \(\dfrac{{{d^2}y}}{{d{x^2}}} = 0\)

(iii) Determine the nature of the stationary points

(iv) Sketch the curve in the domain \( - 1 \le x \le 3\)

The adjacent diagram shows the sketch of \(y = {x^3} + 4{x^2} - 8\). The co-ordinates of the local maximum are?

The adjacent diagram shows the sketch of \(y = 3{x^4} + 4{x^3} - 12{x^2} - 24x\). The co ordinates of the local maximum are ?

The adjacent diagram shows the sketch of \(y = - {x^3} - {x^2} + 4x\). The co-ordinates of the point of inflection are?

The gradient function of a curve is given by \(f^{\prime}(x)=3(x+1)(x-3)\) and the curve \(y=f(x)\) passes through the point \((0,12)\)

i) Find the equation of the curve \(y=f(x)\).

ii) Sketch the curve \(y=f(x)\), clearly labeling turning points and the \(y\) intercept.

iii) Assuming there is a point of inflexion at \(x=1\), determine the values of \(x\) for which the curve is concave up?