The curve with equation \(y = e^{2x}(\sin x + 3 \cos x)\) has a stationary point in the interval \(0 \leq x \leq \pi\).
(a) Find the \(x\)-coordinate of this point, giving your answer correct to 2 decimal places.
(b) Determine whether the stationary point is a maximum or a minimum.
The curve with equation \(y = \frac{e^{-2x}}{1-x^2}\) has a stationary point in the interval \(-1 < x < 1\). Find \(\frac{dy}{dx}\) and hence find the \(x\)-coordinate of this stationary point, giving the answer correct to 3 decimal places.
A curve has equation \(y = \frac{e^{3x}}{\tan \frac{1}{2}x}\). Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\). Give your answers correct to 3 decimal places.
The curve with equation \(y = \frac{2 - \\sin x}{\\cos x}\) has one stationary point in the interval \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\).
(i) Find the exact coordinates of this point.
(ii) Determine whether this point is a maximum or a minimum point.
The equation of a curve is \(y = \frac{\sin x}{1 + \cos x}\), for \(-\pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.
The curve with equation \(y = \frac{{(\ln x)^2}}{x}\) has two stationary points. Find the exact values of the coordinates of these points.
The curve with equation \(y = \\sin x \\cos 2x\) has one stationary point in the interval \(0 < x < \frac{1}{2} \pi\). Find the x-coordinate of this point, giving your answer correct to 3 significant figures.
The curve with equation \(y = \frac{e^{2x}}{4 + e^{3x}}\) has one stationary point. Find the exact values of the coordinates of this point.
A curve has equation \(y = \cos x \cos 2x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac{1}{2}\pi\), giving your answer correct to 3 significant figures.
The equation of a curve is
\(y = 3 \cos 2x + 7 \sin x + 2\).
Find the \(x\)-coordinates of the stationary points in the interval \(0 \leq x \leq \pi\). Give each answer correct to 3 significant figures.
The equation of a curve is \(y = \sin x \sin 2x\). The curve has a stationary point in the interval \(0 < x < \frac{1}{2}\pi\).
Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
The equation of a curve is \(y = \frac{1+x}{1+2x}\) for \(x > -\frac{1}{2}\). Show that the gradient of the curve is always negative.
The curve with equation \(y = \frac{e^{2x}}{x^3}\) has one stationary point.
The equation of a curve is \(y = 3 \sin x + 4 \cos^3 x\).
(i) Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
(ii) Determine the nature of the stationary point in this interval for which \(x\) is least.
The equation of a curve is \(y = \frac{e^{2x}}{1 + e^{2x}}\). Show that the gradient of the curve at the point for which \(x = \ln 3\) is \(\frac{9}{50}\).
The curve \(y = \frac{\ln x}{x^3}\) has one stationary point. Find the x-coordinate of this point.
A curve has equation \(y = e^{-3x} \tan x\). Find the x-coordinates of the stationary points on the curve in the interval \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\). Give your answers correct to 3 decimal places.
The curve \(y = \frac{e^x}{\cos x}\), for \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\), has one stationary point. Find the \(x\)-coordinate of this point.
The curve with equation \(y = e^{-x} \sin x\) has one stationary point for which \(0 \leq x \leq \pi\).
(i) Find the \(x\)-coordinate of this point.
(ii) Determine whether this point is a maximum or a minimum point.
The curve with equation \(y = 6e^x - e^{3x}\) has one stationary point.
(i) Find the \(x\)-coordinate of this point.
(ii) Determine whether this point is a maximum or a minimum point.