Question

f(x) = (x \[-\] 5)⁴.

Answer 1

\[\text { Given: } \hspace{0.167em} f\left( x \right) = \left( x - 5 \right)^4 \]

\[ \Rightarrow f'\left( x \right) = 4 \left( x - 5 \right)^3 \]

\[\text { For a local maximum or a local minimum, we must have }\]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 4 \left( x - 5 \right)^3 = 0\]

\[ \Rightarrow x = 5\]

Since f '(x) changes from negative to positive when x increases through 5, x = 5 is the point of local minima.
The local minimum value of  f (x) at x = 5 is given by \[\left( 5 - 5 \right)^4 = 0\] .