Increasing & Decreasing Functions Flashcards

All 6 cards in this deck

What is true of $\frac{dy}{dx}$ where a function is increasing?

$\frac{dy}{dx} > 0$ (the gradient is positive).

What is true of $\frac{dy}{dx}$ where a function is decreasing?

$\frac{dy}{dx} < 0$ (the gradient is negative).

What are the steps to show a function is increasing at a given point?
e.g. show $y = x^2 + 3x$ is increasing at $x = 1$

1. Differentiate: $\frac{dy}{dx} = 2x + 3$
2. Substitute the $x$-value: $\frac{dy}{dx} = 5$
3. State the sign: $5 > 0$, so the function is increasing.

What are the steps to find the range of values of $x$ for which a function is increasing?
e.g. $y = x^2 - 6x$

1. Differentiate: $\frac{dy}{dx} = 2x - 6$
2. Set up $\frac{dy}{dx} > 0$: $2x - 6 > 0$
3. Solve the inequality: $x > 3$

What are the steps to find the values of a constant $k$ for which a function is increasing for all $x$?
e.g. $f'(x) = 3x^2 + kx + 3$

1. Increasing for all $x$ means $f'(x) > 0$ for all $x$
2. As the $x^2$ coefficient is positive, require discriminant $< 0$: $k^2 - 36 < 0$
3. Solve: $-6 < k < 6$

True or false? A single function can be increasing for some values of $x$ and decreasing for others.

True. The sign of $\frac{dy}{dx}$ can change, e.g. a quadratic decreases then increases.