Pancyprian Examinations (Cyprus) · Mathematics

Applications of differentiation

15 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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Find the $x$ -coordinates of the stationary points of $f (x) = x^{3} - 6 x^{2} + 9 x$ .
On which interval is $f (x) = x^{3} - 6 x^{2} + 9 x$ strictly decreasing?
What is the local maximum value of $f (x) = x^{3} - 6 x^{2} + 9 x$ ?
Where is the point of inflection of the graph of $f (x) = x^{3} - 6 x^{2} + 9 x$ ?
Find the greatest and least values of $f (x) = x^{3} - 3 x^{2}$ on the interval $[- 1, 3]$ .
A square sheet of side $12$ cm has a square of side $x$ cm cut from each corner, and the sides are folded up to form an open box. What is the greatest possible volume of the box?
Two positive numbers have sum $20$ . What is the least possible value of the sum of their squares?
Find the equation of the normal to $y = x^{2} + 2 x$ at the point where $x = 1$ .
The radius of a sphere increases at $2$ cm/s. How fast is the volume increasing, in $cm^{3} / s$ , when the radius is $5$ cm?
What is the maximum value of $f (x) = x e^{- x}$ for real $x$ ?
What does Fermat's theorem say about a function $f$ that is differentiable at an interior point $x_{0}$ where it has a local extremum?
On which interval is the graph of $f (x) = x^{4} - 4 x^{3}$ concave down?
For $f (x) = x^{3}$ we have $f^{'} (0) = 0$ . What does this tell us about $f$ at $x = 0$ ?
The function $f (x) = x^{3} + a x^{2} + b x$ has stationary points at $x = 1$ and $x = 3$ . Find $a$ and $b$ .
How many real solutions does the equation $x^{3} - 3 x + 4 = 0$ have?

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