Applications of Derivatives — A-Level Maths practice questions

Question 1

Among rectangles with perimeter $20$, the one with maximum area has dimensions:

Accepted Answer

$A(x) = 10x - x^2$. $A'(x) = 10 - 2x = 0 \Rightarrow x = 5$. The maximum-area rectangle is the square $5 	imes 5$, with area $25$.

Answer

$4 	imes 6$

Answer

$3 	imes 7$

Answer

$5 	imes 5$

Answer

$2 	imes 8$

Question 2

The function $f(x) = x^3 - 3x^2$ is decreasing on:

Accepted Answer

$f'(x) < 0 \Leftrightarrow 0 < x < 2$. So $f$ is decreasing on $(0, 2)$. (Increasing on $(-\infty, 0)$ and $(2, +\infty)$.)

Answer

$\mathbb{R}$

Answer

$(-\infty, 0)$

Answer

$(0, 2)$

Answer

$(2, +\infty)$

Question 3

For the same position $x(t) = t^3 - 6t^2 + 9t$, the acceleration at $t = 1$ is:

Accepted Answer

$a(t) = x''(t) = 6t - 12$. At $t = 1$: $a = -6$ m/s².

Answer

$-6$ m/s²

Answer

$0$ m/s²

Answer

$6$ m/s²

Answer

$9$ m/s²

Question 4

The point of inflection of $f(x) = x^3 - 3x$ is at:

Accepted Answer

$f'(x) = 3x^2 - 3$, $f''(x) = 6x$. $f''$ changes sign at $x = 0$, so the inflection is at $x = 0$.

Answer

$x = -1$

Answer

$x = 0$

Answer

$x = 1$

Answer

$f$ has no inflection

Question 5

The function $f(x) = x^3$ is concave on:

Accepted Answer

$f''(x) = 6x$. $f'' < 0 \Leftrightarrow x < 0$. So $f$ is concave on $(-\infty, 0)$.

Answer

$\mathbb{R}$

Answer

$(-\infty, 0)$

Answer

$(0, +\infty)$

Answer

nowhere

Question 6

The function $f(x) = x^{2}$ is:

Accepted Answer

$f'(x) = 2x$ and $f''(x) = 2 > 0$ for all $x \in \mathbb{R}$.

Since $f''(x) > 0$ everywhere, $f$ is convex (concave up) on all of $\mathbb{R}$ — no inflection point.

Answer

convex on $\mathbb{R}$ ($f''(x) > 0$)

Answer

concave on $\mathbb{R}$

Answer

convex only on $(0,\infty)$

Answer

has an inflection point at $x = 0$

Question 7

A particle's position is $x(t) = t^3 - 6t^2 + 9t$ (in metres, $t$ in seconds). The velocity at $t = 2$ is:

Accepted Answer

$v(t) = 3t^2 - 12t + 9$. At $t = 2$: $v = 12 - 24 + 9 = -3$ m/s (the particle is moving backward).

Answer

$-3$ m/s

Answer

$0$ m/s

Answer

$3$ m/s

Answer

$9$ m/s

Question 8

The function $f(x) = -x^2 + 4x - 1$ has a local maximum at:

Accepted Answer

$f'(x) = -2x + 4 = 0 \Rightarrow x = 2$. $f''(x) = -2 < 0$, confirming a local maximum.

Answer

$x = -2$

Answer

$x = 0$

Answer

$x = 2$

Answer

$x = 4$

Question 9

The function $f(x) = x^{3} - 3x$ is **decreasing** on the interval:

Accepted Answer

$f'(x) = 3x^{2} - 3 = 3(x - 1)(x + 1)$.

$f'(x) < 0$ when $(x-1)(x+1) < 0$, i.e. on $(-1, 1)$.

So $f$ is decreasing on $(-1, 1)$.

Answer

$(-\infty, -1)$

Answer

$(-1, 1)$

Answer

$(1, \infty)$

Answer

$\mathbb{R}$

Question 10

For which value of $a$ does $f(x) = x^3 - 3ax + 1$ have a local minimum at $x = 2$?

Accepted Answer

$f'(x) = 3x^2 - 3a$, so $f'(2) = 12 - 3a = 0 \Rightarrow a = 4$. Check: $f''(x) = 6x$, so $f''(2) = 12 > 0$, confirming a local minimum.

Answer

$a = 1$

Answer

$a = 2$

Answer

$a = 3$

Answer

$a = 4$

Applications of Derivatives practice questions — A-Level Maths

Problems & worked solutions

Problem 1 — Applications of Derivatives

Problem 2 — Applications of Derivatives

Problem 3 — Applications of Derivatives

Problem 4 — Applications of Derivatives

Problem 5 — Applications of Derivatives

Problem 6 — Applications of Derivatives

Problem 7 — Applications of Derivatives

Problem 8 — Applications of Derivatives

Problem 9 — Applications of Derivatives

Problem 10 — Applications of Derivatives