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Use bisection on $f^{\prime }$ when the derivative is cheap and a bracket $f^{\prime }(a)<0<f^{\prime }(b)$ is easy to obtain. For a differentiable convex function, $f^{\prime }$ is nondecreasing, so such a bracket locates a global minimum, allowing a flat minimizing interval. Without convexity, first isolate a region containing the desired local minimum; arbitrary stationary points need not be minima.

Use golden-section search when reliable function values are available but derivatives are unavailable or expensive. It needs a bracket on which the objective is a unimodal function, commonly found by three ordered points whose middle value is below the two outer values. Those inequalities alone do not prove that the objective is a unimodal function. A convex function supplies the needed shape; for a nonconvex objective, isolate an interval on which it is a unimodal function.

Both methods have linear convergence of bracket width. Bisection contracts the width by $1/2$ per new derivative evaluation. Golden-section search contracts it by $(\sqrt{5}-1)/2\approx 0.618$ per new function evaluation after initialization, reusing one earlier value. Choose by evaluation cost and bracket availability as well as this contraction factor: comparing iteration counts alone can misidentify the faster method.

A real function on an interval is unimodal if there is a point $m$ such that it is nondecreasing to the left of $m$ and nonincreasing to the right. This convention describes a single peak and allows flat regions. Some optimization texts instead use the corresponding single-valley convention, so the direction must be specified.

For example, $f(x)=-(x-2{)}^2$ has its peak at $2$. A unimodal density can describe a distribution with one mode, while unimodality of an objective allows interval-elimination search methods. Concavity implies peak unimodality, but unimodality alone does not require the chord inequalities of concavity.