The equation of a curve is \(y = 2 + \sqrt{25 - x^2}\).
Find the coordinates of the point on the curve at which the gradient is \(\frac{4}{3}\).
The equation of a curve is \(y = 54x - (2x - 7)^3\).
(a) Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(b) Find the coordinates of each of the stationary points on the curve.
(c) Determine the nature of each of the stationary points.
The equation of a curve is \(y = (3 - 2x)^3 + 24x\).
(a) Find expressions for \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(b) Find the coordinates of each of the stationary points on the curve.
(c) Determine the nature of each stationary point.
The equation of a curve is \(y = x^3 + x^2 - 8x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).
The line \(y = ax + b\) is a tangent to the curve \(y = 2x^3 - 5x^2 - 3x + c\) at the point \((2, 6)\). Find the values of the constants \(a, b\) and \(c\).
A curve has equation \(y = (2x - 1)^{-1} + 2x\).
(i) Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(ii) Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point.
The curve with equation \(y = x^3 - 2x^2 + 5x\) passes through the origin.
(i) Show that the curve has no stationary points.
(ii) Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
Functions f and g are defined by
\(f(x) = \frac{8}{x-2} + 2\) for \(x > 2\),
Find the set of values of \(x\) satisfying the inequality \(6f'(x) + 2f^{-1}(x) - 5 < 0\).
A curve has equation \(y = \frac{1}{2}x^{\frac{1}{2}} - 4x^{\frac{3}{2}} + 8x\).
(i) Find the \(x\)-coordinates of the stationary points.
(ii) Find \(\frac{d^2y}{dx^2}\).
(iii) Find, showing all necessary working, the nature of each stationary point.
A curve is such that \(\frac{dy}{dx} = -x^2 + 5x - 4\).
(i) Find the \(x\)-coordinate of each of the stationary points of the curve.
(ii) Obtain an expression for \(\frac{d^2y}{dx^2}\) and hence or otherwise find the nature of each of the stationary points.
Find the coordinates of the minimum point of the curve \(y = \frac{9}{4}x^2 - 12x + 18\).
The equation of a curve is \(y = 8\sqrt{x} - 2x\).
The function \(f\) is defined for \(x \geq 0\) by \(f(x) = (4x + 1)^{\frac{3}{2}}\).
(i) Find \(f'(x)\) and \(f''(x)\).
The first, second and third terms of a geometric progression are respectively \(f(2)\), \(f'(2)\) and \(kf''(2)\).
(ii) Find the value of the constant \(k\).
A curve has equation \(y = 8x + (2x - 1)^{-1}\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.
The function f is defined by \(f(x) = 2x + (x + 1)^{-2}\) for \(x > -1\).
Find \(f'(x)\) and \(f''(x)\) and hence verify that the function f has a minimum value at \(x = 0\).
A curve has equation \(y = \frac{8}{x} + 2x\).
(i) Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).
(ii) Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.
Variables u, x and y are such that \(u = 2x(y - x)\) and \(x + 3y = 12\). Express u in terms of x and hence find the stationary value of u.
The equation of a curve is \(y = x^3 + px^2\), where \(p\) is a positive constant.
(i) Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\).
(ii) Find the nature of each of the stationary points.
Another curve has equation \(y = x^3 + px^2 + px\).
(iii) Find the set of values of \(p\) for which this curve has no stationary points.
A function \(f\) is such that \(f(x) = \frac{15}{2x+3}\) for \(0 \leq x \leq 6\).
Find an expression for \(f'(x)\) and use your result to explain why \(f\) has an inverse.
A curve has equation \(y = \frac{k^2}{x+2} + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.