The equation of a curve is \(y = \frac{2}{\sqrt{5x - 6}}\).
Find the gradient of the curve at the point where \(x = 2\).
A curve has equation \(y = ax^{\frac{1}{2}} - 2x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9.
Find the \(y\)-coordinate of \(P\).
The non-zero variables x, y and u are such that u = x2y. Given that y + 3x = 9, find the stationary value of u and determine whether this is a maximum or a minimum value.
A curve has equation \(y = f(x)\) and is such that \(f'(x) = 3x^{\frac{1}{2}} + 3x^{-\frac{1}{2}} - 10\).
(i) By using the substitution \(u = x^{\frac{1}{2}}\), or otherwise, find the values of \(x\) for which the curve \(y = f(x)\) has stationary points.
(ii) Find \(f''(x)\) and hence, or otherwise, determine the nature of each stationary point.
A curve is such that \(\frac{dy}{dx} = 2(3x + 4)^{\frac{3}{2}} - 6x - 8\).
(i) Find \(\frac{d^2y}{dx^2}\).
(ii) Verify that the curve has a stationary point when \(x = -1\) and determine its nature.
A curve has equation \(y = 2x + \frac{1}{(x-1)^2}\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.
It is given that a curve has equation \(y = f(x)\), where \(f(x) = x^3 - 2x^2 + x\).
(i) Find the set of values of \(x\) for which the gradient of the curve is less than 5.
(ii) Find the values of \(f(x)\) at the two stationary points on the curve and determine the nature of each stationary point.
A curve has equation \(y = 3x^3 - 6x^2 + 4x + 2\). Show that the gradient of the curve is never negative.
Function g is defined by
\(g : x \mapsto 2(x-1)^3 + 8, \quad x > 1\).
Obtain an expression for \(g'(x)\) and use your answer to explain why \(g\) has an inverse.
The variables x, y and z can take only positive values and are such that
\(z = 3x + 2y\) and \(xy = 600\).
(i) Show that \(z = 3x + \frac{1200}{x}\).
(ii) Find the stationary value of \(z\) and determine its nature.
A curve has equation \(y = \frac{1}{x-3} + x\).
(i) Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(ii) Find the coordinates of the maximum point \(A\) and the minimum point \(B\) on the curve.
The equation of a curve is \(y = \frac{9}{2-x}\).
Find an expression for \(\frac{dy}{dx}\) and determine, with a reason, whether the curve has any stationary points.
The equation of a curve is \(y = 3x + 1 - 4(3x + 1)^{\frac{1}{2}}\) for \(x > -\frac{1}{3}\).
(a) Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(b) Find the coordinates of the stationary point of the curve and determine its nature.
The equation of a curve is \(y = (2x - 3)^3 - 6x\).
(i) Express \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) in terms of \(x\).
(ii) Find the \(x\)-coordinates of the two stationary points and determine the nature of each stationary point.
A curve has equation \(y = \frac{k}{x}\). Given that the gradient of the curve is \(-3\) when \(x = 2\), find the value of the constant \(k\).
Find the gradient of the curve \(y = \frac{12}{x^2 - 4x}\) at the point where \(x = 3\).
A curve has equation \(y = x^2 + \frac{2}{x}\).
(i) Write down expressions for \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(ii) Find the coordinates of the stationary point on the curve and determine its nature.
A curve has equation \(y = x^3 + 3x^2 - 9x + k\), where \(k\) is a constant.
(i) Write down an expression for \(\frac{dy}{dx}\).
(ii) Find the \(x\)-coordinates of the two stationary points on the curve.
(iii) Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.
The function \(f\) is defined by \(f(x) = x^2 + \frac{k}{x} + 2\) for \(x > 0\).
(a) Given that the curve with equation \(y = f(x)\) has a stationary point when \(x = 2\), find \(k\).
(b) Determine the nature of the stationary point.
(c) Given that this is the only stationary point of the curve, find the range of \(f\).
The gradient of a curve is given by \(\frac{dy}{dx} = 6(3x - 5)^3 - kx^2\), where \(k\) is a constant. The curve has a stationary point at \((2, -3.5)\).
(a) Find the value of \(k\).
(c) Find \(\frac{d^2y}{dx^2}\).
(d) Determine the nature of the stationary point at \((2, -3.5)\).