Does the parent secant function have points of inflection?

Question

just some trig things im confused about

Answer 1

sec'(x) = sec(x)tan(x)

sec''(x) = [sec(x)tan(x)]tan(x) + sec(x)sec²(x)

= sec(x)tan²(x) + sec(x)sec²(x)

= sec(x)[tan²(x) + sec²(x)]

= sec(x)[tan²(x) + (1+tan²(x))]

= sec(x)[1 + 2tan²(x)] = 0

1) sec(x) = 0 → No soln.

2) 1 + 2tan²(x) = 0

2tan²(x) = -1

tan²(x) = -½

tan(x) = √(-½) → No soln.

Answer: No

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I know it seems like there should be points of inflection at the infinite discontinuities, but the function must be differentiable on the interval containing the discontinuities.

Answer 2

i think none