uppose f (x) has a local minimum at x = c and define g(x) = −f (x). Show that g(x) has a local maximum at x = c. (Hint: Use the definition of a local maximum.)

Answer 1

By the definition of local minimum:

There exists δ>0 such that for all x∈(c-δ, c+δ), f(x) ≥ f(c).

Substitute f(x) = -g(x)

There exists δ>0 such that for all x∈(c-δ, c+δ), -g(x) ≥ -g(c).

Multiply the expression by -1:

There exists δ>0 such that for all x∈(c-δ, c+δ), g(x) ≤ g(c).

The above is the definition of g(x) having local maximum at c.