L2 Calculus
Question Bank
Use calculus to verify that $x = 2$ is a stationary point of the curve $y = x^3 - 6x^2 + 12x - 5$.
Use calculus to determine whether $x = 4$ is a stationary point of the curve $y = x^2 - 6x + 5$.
Find the coordinates and nature of the stationary points of the curve $y = x^3 - 3x^2 - 9x + 5$.
The graph of a function $f(x) = x^3 + bx^2 + 5$ has a turning point when $x=4$. Find the value of $b$.
Find the coordinates and nature of the stationary points of the curve $y = x^4 - 8x^3 + 16x^2$.
Find the coordinates and nature of the stationary points of the curve $y = x^4 - 6x^3 + 9x^2 + 2$.
A company's weekly profit (in thousands of dollars) is modelled by $P(x) = -2x^2 + 16x - 10$, where $x$ is the number of units produced (in hundreds).
The company manager claims that producing 400 units per week (i.e., $x = 4$) will give a stationary point for profit.
Use calculus to verify whether the manager is correct.
The curve $y = x^3 - 6x^2 + kx + 5$ has a stationary point at $x = 3$, where $k$ is a constant.
Use calculus to find the value of $k$.
Find the coordinates and nature of the stationary points of the curve $y = x^4 - 8x^3 + 22x^2 - 24x + 12$.
Use calculus to find the quadratic function $f(x)$ that has the following properties:
- a turning point when $x = 3$
- a gradient of -8 when $x = 1$
- a value of -5 when $x = 5$
The function $f(x) = x^4 + ax^3 + bx^2 + 3$ has turning points when $x = 1$ and $x = -2$. Find the values of $a$ and $b$.
Use calculus to prove that the graph of the quartic function $y = x^2(x-4)^2$ has a local maximum when $x=2$.
The graph of the function $y = x^3 - 3x^2 + kx + 10$ has a turning point at $x = 4$. Find the coordinates of both turning points and determine their nature.
A cubic function, $f(x) = ax^3 + bx^2 + cx + d$, has turning points that occur at $x = -2$ and $x = 4$. Find an expression for $b$ in terms of $a$.