For the curve $y = 4x^2 + 6x$, find the stationary point(s) and determine their nature using the second derivative.
$\left(-\dfrac{3}{4}, -\dfrac{9}{4}\right)$ min
For the curve $y = x(x^2 - 4x - 3)$, find the stationary point(s) and determine their nature using the second derivative.
$\left(3,-18\right)$ min, $\left(-\dfrac{1}{3},\dfrac{14}{27}\right)$ max
For the curve $y = x - 3\sqrt{x}$, find the stationary point(s) and determine their nature using the second derivative.
$\left(\dfrac{9}{4}, -\dfrac{9}{4}\right)$ min
For the curve $y = x^4 - 12x^2$, find the stationary point(s) and determine their nature using the second derivative.
$(0,0)$ max, $\left(\pm\sqrt{6},-36\right)$ min
For the curve $y = 2x^3 - 15x^2 + 24x + 6$, find the stationary point(s) and determine their nature using the second derivative.
$(1,17)$ max, $(4,-10)$ min
Find the coordinates of the minimum point of the curve $y = x^2(2-x)$.
$(0,0)$
For the curve $y = x^4 + 3x^3 - 5x^2 - 3x + 1$, find the stationary point(s) and determine their nature using the second derivative.
$\left(-\dfrac{1}{4},\dfrac{357}{256}\right)$ max, $(-3,-35)$ and $(1,-3)$ min
Show that the curve $y = x^3 + 3x^2 + 6x - 4$ has no stationary points.
Discriminant of gradient is less than zero
The total surface area of a cylinder of radius $r$ and height $h$ is $150\pi$. Find the maximum volume of the cylinder.
$r = 5$, $h = 10$ and $V = 250\pi$
A cuboid has dimensions $x$ by $x$ by $y$. Given that the surface area of the cuboid is $k$, show that the maximum volume of the cuboid occurs when $y = x$.
max when $x = \sqrt{\dfrac{6}{k}}$
Prove that the graph of $y = 3x^2 + 5x - 2$ is always concave up.
$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 6$ which is always greater than $0$
Prove that the graph of $y = (5-x)(2+x)$ is always concave down.
$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = -2$ which is always less than $0$
Find the range of values of $x$ for which $y = x^3 - 2x^2$ is concave up.
$x > \dfrac{2}{3}$
Find the range of values of $x$ for which $$y = x^4 + 6x^3 - 10x^2 + 5x - 1$$ is concave down.
$-5 < x < 0.5$
Find the coordinates of the stationary point on the curve $y = x^3 - 3x^2 + 3x + 1$, and determine its nature.
$(1,2)$, point of inflection
Find the coordinates of the stationary point on the curve $y = 4-12x + 6x^2 - x^3$, and determine its nature.
$(2,-4)$, point of inflection
Find the coordinates of the stationary point on the curve $y = (x^2+1)(x^2-1)$, and determine its nature.
$(0,-1)$, min
Find the coordinates of the point of inflection on the curve $y = x^3 - 6x^2 + 5x - 1$ and show that it is not a stationary point.
$(2, -7)$, gradient is not zero
The curve $y = x^4 - 8x^2 + 12$ has two points of inflection. Find their coordinates and show that they are not stationary points.
$\left(\pm\dfrac{2\sqrt{3}}{3}, \dfrac{10}{9}\right)$, gradient is not zero
The curve with equation $y = x^4 + 4x^3 + kx^2$ is always concave up. Find the possible values for $k$.
$k > \dfrac{2}{3}$