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Maximize and Minimize are the ± peak point of slope. You can find it yourself by taking derivative of corresponding equation. Solving it should give you the minimum or maximum point of that equation.

As you can see here, minimize and maximize functions are different.
Reason for this to demonstrate the difference between roots(), FindRoot() and maple solve functions. NOT for there are difference in real world, just to show solvers.
While they are both solvers, they have some fundamental differences. Some of results can be obtained by both, some by only one of them. Sometimes results are wrong sometimes correct.
By checking the sheets below don't get the idea any of mentioned one of solving function is superior to others. All of them have their pros and cons.
Plot the functions, do some hand calculation for error check and find best suitable one for your project.

x f k minimize x f x diff x k roots :

x f k maximize result x f x diff x k ≡ FindRoot : x k Clear result 31 line :

x f maple_maxmin x temp x f x diff : x temp x solve maple 21 line :

h 600 mm * : b 300 mm * : L 10 m * : E 470030 sqrt * MPa * : W 10 kN m / * : 15 mat

x w W L / x * :

x V W 2 / L 3 / x 2^L / - (* :

x M W 6 / x L * x 3^L / - (* :

I b h 3^* 12 / :

x y W 120 E * I * / x 5^L / (- 10 L * 3 / x 3^* + 7 L 3^3 / * x * - (* :

x.L 0 m * (L 0.3 m * range :

x.L m / x.L w s 2 ^ kg / * augment x.L m / x.L M vectorize kN m * / augment

No result, roots() can't solve if units are not eliminated x M 5 minimize #

x M.unit W 6 / x m * (L * x m * (3^L / - (* :

correct with unit eliminated equation x M.unit 5 minimize 5.7735

correct with original equation x M 5 m * maximize 5.7735 m *

could be considered correct x M maple_maxmin 5.7735 m * 5.7735 m * - 21 mat

x.L m / x.L y m / vectorize augment

No result, roots() can't solve if units are not eliminated x y 5 minimize #

x y.unit W 120 E * I * / x 5^m 5^* L / (- 10 L * 3 / x 3^* m 3^* + 7 L 3^3 / * x * m * - (* :

guess value made solve wrong x y.unit 1 minimize 13.1541

correct with unit eliminated x y.unit 1 minimize 13.1541

not correct x y 1 m * maximize 5.241 m *

also not correct x y.unit 2 m * maximize 5.1927 m *

this contains correct answer along with useless results x y maple_maxmin 13.1541 m * 13.1541 m * - 5.1933 m * 5.1933 m * - 41 mat