Applications of derivatives

Tangent and normal lines

For a tangent drawn from a point off the curve, let the point of tangency be $(t,\ f(t))$, write the tangent line and require it to pass through the given point.

Increase, decrease and local extrema

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

Example: for $f(x) = xe^{-x}$, $f'(x) = (1 - x)e^{-x}$. Since $e^{-x} > 0$, the sign is that of $1 - x$, so $f$ has a local maximum $\frac{1}{e}$ at $x = 1$.

Concavity, inflection and curve sketching

The graph is concave up where $f''(x) > 0$ and concave down where $f''(x) < 0$. A point where the concavity changes is a point of inflection.

To sketch a graph, combine the signs of $f'$ and $f''$ in one table, then check the behavior as $x \to \pm\infty$ (asymptotes) and the intercepts.

Equations, inequalities and optimization

Counting real roots: the number of real solutions of $f(x) = a$ is the number of intersections of $y = f(x)$ with the line $y = a$. Separate the constant and use the graph.

Proving inequalities: let $f(x)$ = (left side) − (right side) and show $f(x) > 0$ by studying its increase and decrease, using $f''$ if the sign of $f'$ is not clear.

Maximum and minimum: on a closed interval, compare the local extrema with the values at the endpoints. In geometry problems, express the quantity in one variable and mind its allowed range.