Applications of derivatives: practice problems

Tables of signs and extrema

$f(x)$ increases where $f'(x) > 0$ and decreases where $f'(x) < 0$. A local maximum occurs where $f'$ changes from positive to negative, and a local minimum where it changes from negative to positive.

Example: for $f(x) = x^3 - 3x$, $f'(x) = 3(x + 1)(x - 1)$, so

The local maximum is $2$ at $x = -1$ and the local minimum is $-2$ at $x = 1$.

Maximum and minimum on an interval

On a closed interval $a \leq x \leq b$, compare the local extreme values with the values at the endpoints. A table of signs covering the whole interval avoids mistakes.

In word problems, first determine the allowed range of the variable (for a box of height $x$, for example, $0 < x < \frac{a}{2}$).

Counting real roots

The number of real solutions of $f(x) = k$ equals the number of intersections of $y = f(x)$ with the horizontal line $y = k$. Move the constant to one side and compare $k$ with the local extreme values.

For a cubic with local maximum $M$ and local minimum $m$: three roots if $m < k < M$, two if $k = m$ or $k = M$, and one otherwise.

A cubic $f(x)$ has local extrema exactly when $f'(x) = 0$ has two distinct real roots, i.e. its discriminant is positive.

Inequalities

To show $f(x) \geq 0$ for all $x \geq 0$, show that the minimum of $f$ on $x \geq 0$ is at least $0$. If $f$ contains a constant $k$, find the values of $k$ for which this minimum is non-negative.