Monotonicity of a function
A graph tells you at a glance whether a function goes up or down. But when you don't have the drawing —only the formula for a profit, a cost or an audience— you need another way of knowing. The derivative provides it: its sign, at each point, tells you whether the function is increasing or decreasing right there.
The sign of $f'$ decides the monotonicity
Given the function $f(x) = 2x^3 - 6x$, we want to know how it behaves near the origin. Calculate the gradient of the tangent line (the derivative) at the points…
Given the function $f(x) = 2x^3 - 6x$, we want to know how it behaves near the origin. Calculate the gradient of the tangent line (the derivative) at the points $x = -2$, $x = 0$ and $x = 2$. Based solely on the sign of these results, state whether the function is increasing or decreasing at each of these points.
We know that a function $g(x)$ has derivative $g'(x) = x - 4$. Without needing to know the expression of $g(x)$, determine for which values of $x$ the original function…
We know that a function $g(x)$ has derivative $g'(x) = x - 4$. Without needing to know the expression of $g(x)$, determine for which values of $x$ the original function $g(x)$ is increasing and for which it is decreasing. Justify your answer using the concept of derivative.
A linear function always increases or always decreases
Check whether the linear function $f(x) = 5x - 3$ is always increasing across its entire domain. To do so, calculate its derivative and explain how the result obtained…
Check whether the linear function $f(x) = 5x - 3$ is always increasing across its entire domain. To do so, calculate its derivative and explain how the result obtained relates to the gradient of the line.
Studying monotonicity at a specific point
When a function is not linear, the sign of $f'$ changes depending on the point. To find out what happens at $x_0$ it is enough to calculate $f'(x_0)$: there is no need to solve the equation $f'(x)=0$ or to study the whole function.
The height of a basketball (in metres) when thrown is modelled by the function $h(t) = -5t^2 + 6t + 2$, where $t$ is the time in seconds. 1. Calculate the instantaneous…
The height of a basketball (in metres) when thrown is modelled by the function $h(t) = -5t^2 + 6t + 2$, where $t$ is the time in seconds.
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Calculate the instantaneous velocity of the ball by differentiating the height function.
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Determine the time interval in which the ball is rising (increase of the height function).
Study the monotonicity of the rational function $f(x) = \dfrac{x^2 + 9}{x}$. This type of function is often used to model average production cost in Social Sciences.…
Study the monotonicity of the rational function $f(x) = \dfrac{x^2 + 9}{x}$. This type of function is often used to model average production cost in Social Sciences. State, with justification, the intervals where the average cost is decreasing and those where it is increasing, bearing in mind that in this context only values $x > 0$ make sense.
The figure shows the graph of $f$ for $x > 0$; you can use it to check your result.
Monotonicity on an interval: increasing profit
A market study indicates that the daily profit of a small textile company (in hundreds of euros) is given by the function $B(t) = -t^2 + 10t - 16$, where $t$ is the time…
A market study indicates that the daily profit of a small textile company (in hundreds of euros) is given by the function $B(t) = -t^2 + 10t - 16$, where $t$ is the time in hours since opening. Use the derivative to find in which time interval the company's profit is increasing.
A streaming service company has found that the number of new daily subscribers, in thousands, follows the function $S(t) = -t^3 + 12t^2 + 45t + 100$, where $t$…
A streaming service company has found that the number of new daily subscribers, in thousands, follows the function $S(t) = -t^3 + 12t^2 + 45t + 100$, where $t$ represents the time in months since the start of an advertising campaign ($0 \leq t \leq 15$).
a) Determine the time intervals in which the growth of new subscribers is positive (the function is increasing).
b) From which month onwards does the trend change and the number of new daily subscribers starts to decrease? Justify the answer by studying the first derivative.