Just a handful of questions to make sure you're doing integration and differentiation more or less right...
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Where is/are the turning point(s) of $y = x^3 - 12x^2 + 45x - 20$?
Minimum at $(0, -20)$
Maximum at $(3, 34)$, minimum at $(5, -68)$
Stationary point of inflection at $(4, 32)$
Minimum at $(3, 34)$, maximum at $(5, - 68)$
Maximum at $(0, -20)$
The perimeter of a running track is 400 metres, and consists of two semi-circles of radius $r$ connected by two straights of length $l$. What is the maximum area enclosed by the running track, in m$^2$?
The largest possible shape with a fixed perimeter is a circle! It turns out that $l = 0$.
Find the area enclosed by the curve $ y = (x+3)(x-2)(x-5)$ and the $x$-axis.
The curve $C$ has the equation $y = x^2 - 3x - 18$, and crosses the $x$-axis at points $P$ and $Q$. The lines $l_1$ and $l_2$ are tangent to the curve at $P$ and $Q$, and meet at the point $R$. What is the area of triangle $PQR$?
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