Curve Sketching

Four shapes cover the local picture, one for each pairing of the two signs. With $f' > 0$ and $f'' > 0$ the curve rises and steepens. With $f' > 0$ and $f'' < 0$ it still rises, but with the slope decreasing, so the climb gets shallower without having to level off. With $f' < 0$ and $f'' > 0$ it falls with the slope increasing from below, so each step down is shallower than the last. With $f' < 0$ and $f'' < 0$ it falls ever more steeply. Direction comes from $f'$ and bend comes from $f''$.

A sign chart keeps it organised. Mark every $x$ where $f'$ is zero or undefined on one number line and every $x$ where $f''$ is zero or undefined on a second, test a point inside each interval, then read the shape off the pair of rows. Those rows describe the shape between the marked points and no more: $f' > 0$ with $f'' < 0$ fits $\ln x$, $\sqrt{x}$ and $1 - e^{-x}$ alike, three graphs that look nothing like each other, so intercepts, asymptotes and end behaviour limits are what pin the actual picture down. The chart is scratch work; the credit is in the sentence it lets you write, such as $f$ has a relative maximum at $x = 3$ because $f$ is continuous at $x = 3$ and $f'$ changes from positive to negative there. At a marked point where $f'$ is undefined, confirm that $f$ is continuous before applying that test.