A curve is such that its gradient at the point \((x,y)\) is given by \((5x-2)^{\frac13}\). The curve passes through the point \(\left(2,\frac{32}{5}\right)\).

Find the coordinates of the stationary point on the curve.

DO NOT USE A CALCULATOR IN THIS QUESTION. (a) Given that \(x-3\) and \(x+1\) are both factors of \(2 x^{3}-3 x^{2}-8 x-3\), solve the equation \(2 x^{3}-3 x^{2}-8 x-3=0\). (b) The polynomial \(\mathrm{p}(x)=x^{3}+a x^{2}+b x+c\), where \(a, b\) and \(c\) are constants, has remainder -5 when divided by \(x-1\). The curve \(y=\mathrm{p}(x)\) has stationary points at \(x=\frac{4}{3}\) and \(x=2\). (i) Find the values of \(a, b\) and \(c\). (ii) Hence use the second derivative test to show that the stationary point at \(x=2\) is a minimum.

A curve has equation

\(y=32x^2+\frac{1}{8x^2},\qquad x\ne0.\)

(a) Find the coordinates of the stationary points of the curve.

(b) These stationary points have the same nature. Use the second derivative test to determine whether they are maximum points or minimum points.

The function \(f\) is defined by

\(f(x)=(2x+1)(3x-2)^2\).

(a) Show that \(f'(x)\) can be written in the form \(2(3x-2)(px+q)\), where \(p\) and \(q\) are integers to be found.

(b) Find the coordinates of the stationary points of the graph of \(y=f(x)\).

(c) Sketch the graph of \(y=f(x)\), showing clearly the intercepts with the axes and the stationary points.

(d) Find the set of values of the constant \(k\) for which the equation \(f(x)=k\) has 3 distinct real roots.

The polynomial \(\mathrm q(x)\) is given by

\(\mathrm q(x)=-\frac13(2x-1)(x+3)^2.\)

(a) Find the \(x\)-coordinates of the stationary points on the curve \(y=\mathrm q(x)\).

(b) On the axes, sketch the graph of \(y=\mathrm q(x)\), stating the intercepts with the coordinate axes.

(c) Find the values of \(k\) such that \(\mathrm q(x)=k\) has exactly one solution.

Do not use a calculator in this question.

A curve has equation

\(y=(2-\sqrt3)x^2+x-1.\)

The \(x\)-coordinate of a point \(A\) on the curve is

\(\frac{\sqrt3+1}{2-\sqrt3}.\)

(a) Show that the coordinates of \(A\) can be written in the form \((p+q\sqrt3,r+s\sqrt3)\), where \(p,q,r\) and \(s\) are integers.

(b) Find the \(x\)-coordinate of the stationary point on the curve, giving your answer in the form \(a+b\sqrt3\), where \(a\) and \(b\) are rational numbers.

A curve has equation

\(y=(3+\sqrt5)x^2-8\sqrt5\,x+60.\)

(a) Find the \(x\)-coordinate of the stationary point of the curve in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

(b) Find the \(y\)-coordinate of this stationary point in the form \(c\sqrt5\), where \(c\) is an integer.

(a) Find the \(x\)-coordinates of the stationary points on the curve

\(y=3\ln x+x^2-7x,\)

where \(x\gt 0\).

(b) Determine the nature of each of these stationary points.

\(y=(x^2-1)\sqrt{5x+2}.\)

(a) Show that

\(\frac{dy}{dx}=\frac{Ax^2+Bx+C}{2\sqrt{5x+2}},\)

where \(A\), \(B\) and \(C\) are integers to be found.

(b) Find the coordinates of the stationary point of the curve for \(x\gt0\). Give your answer correct to 2 significant figures.

(c) Determine the nature of this stationary point.

(a) Given that

\(y=x\sqrt{x+2},\)

show that

\(\frac{dy}{dx}=\frac{Ax+B}{2\sqrt{x+2}},\)

where \(A\) and \(B\) are constants.

(b) Find the exact coordinates of the stationary point of the curve \(y=x\sqrt{x+2}\).

(c) Determine the nature of this stationary point.

It is given that

\(y=(2x-1)\sqrt{4x+3}.\)

(a) Show that

\(\frac{dy}{dx}=\frac{4(Ax+B)}{\sqrt{4x+3}},\)

where \(A\) and \(B\) are constants to be found.

(b) Find the \(x\)-coordinate of the stationary point and determine its nature.

Find the coordinates of the stationary point of the curve

\(y=\frac{x+2}{\sqrt{2x-1}}.\)

The equation of a curve is

\(y=x^2\sqrt{3+x}\)

for \(x\geq-3\).

(i) Find \(\dfrac{dy}{dx}\).

(ii) Find the equation of the tangent to the curve \(y=x^2\sqrt{3+x}\) at the point where \(x=1\).

(iii) Find the coordinates of the turning points of the curve \(y=x^2\sqrt{3+x}\).

Show that the curve \(y=(3x^2+8)^{5/3}\) has only one stationary point. Find the coordinates of this stationary point and determine its nature.