Finding maxima and minima of a continuous time signal by using frequency method?
Global maximum

Representation of a signal from DFT components is:

$$f[n] = \frac{1}{N} \sum_{k=0}^{N-1} F[k] e^{i 2\pi k n /N}$$

Let $\theta = 2\pi n /N$

$$f[N\theta/(2\pi)] = \frac{1}{N} \sum_{k=0}^{N-1} F[k] e^{i k \theta}$$

This is a trigonometric polynomial with coefficients $F[k]/N$. Finding the maximum is the same as for a normal polynomial, at the maximum the derivative is 0. Thus you calculate the derivative polynomial,

$$\frac{1}{N} \sum_{k=0}^{N-1} i k F[k] e^{i k \theta}$$

Use root finding to obtain $N-1$ possible values of $\theta$ (the roots), substitute them back into the original polynomial and choose the one that corresponds to the maximum. Then $n$ can be found as $n = N\theta/(2\pi)$.

Global minimum

Same as the above except you choose the value of the original polynomial that is a minimum

Local maxima and minima

Find the $N-1$ roots of the polynomial as before. Discard any points which do not correspond to derivative of zero. The remaining points are local maxima or minima. At the local maxima the second derivative is negative, at local minima it is positive.