2. Let f: DR where DCR is open, and suppose that f € C² (D). Let xo E D andsuppose that B(xo, e) C D for some € > 0 (this is guaranteed by the fact that D is open).(a) Prove that8² fəxkxjand that8² fəxjxk-(xo)-(xo)==1lim lim (f(xo + tek + sej)-f(xo + tek)-f(xo + sej) +t-0s 0 tsf(xa))]1lim lim (f(xo + se; + tek) - f(xo + sej) - f(xo + tex) + f(xo+ f(xo))].8-0 t-0 ts(Hint: If you can prove one of the identities, the other one follows by the identicalargument, so you only need to prove one of them.)

Given that f:D→ℝ, where D⊆ℝn is open and f∈C2D. Let x0∈D. We know that: ∂f∂xix0=limh→0fx0+hei-fx0h.…

2. Let f: DR where DCR is open, and suppose that f € C²(D). Let xo E D and suppose that B(xo, e) CD for some > 0 (this is guaranteed by the fact that D is open). (a) Prove that 1 8² f (xo) = lim lim (f(xo + tek + sej) — f(xo + tek) — f(xo + sej) + f(xo) Әхкдх ј + f(x))] t-0 8-0 ts and that 8² f дхјдхк 1 -(xo) = lim lim (f(xo + sej + tek) — ƒ (xo + se;) − ƒ (xo + ter) + f(xo); -f(xo))]. 8-0 t-0 ts