Local Max and Min of a graph...

Question

The function 𝑓(𝑥)= 5𝑥 + 9𝑥^-1 has one local minimum and one local maximum.

This function has a local maximum at 𝑥=   

with value   

and a local minimum at 𝑥=   

with value   

Answer 1

f(x) = 5x + 9/x; f'(x) = 5 - 9/x² = 0; x = ±3/√5

f''(x) = 18/x³; f''(3/√5) = 10√5/3 > 0. So, at x = 3/√5 f(x) has local minimum f(3/√5) = 6√5

f''(-3/√5) = -10√5/3 < 0. So, at x = -3/√5 f(x) has local minimum f(-3/√5) = -6√5.