Let 𝑓(𝑥, 𝑦) = ൜0, 𝑥 𝑦 = 01, 𝑥 𝑦 ≠ 0(a) Find the limit of 𝑓 as (𝑥, 𝑦) approaches (0, 0) along the line 𝑦 = 𝑥.(b) Prove that 𝑓 is not continuous at the origin.(c) Show that both partial derivatives 𝑓௫ and 𝑓௬ exist at the origin

Determine the continuity of the function 𝑓𝑥=𝑥3-𝑥

(x)=⎧⎩⎨⎪⎪⎪⎪|x+1|,1,x+1√−1x,x<0x=0x>0.𝑓(𝑥)={|𝑥+1|,𝑥<01,𝑥=0𝑥+1−1𝑥,𝑥>0. Which of the following is/are incorrect?Question 1Answera.f(x)𝑓(𝑥) is continuous at x=0𝑥=0.b.f(x)𝑓(𝑥) is continuous for all real numbers x𝑥. c.f(x)𝑓(𝑥) is differentiable at x=0𝑥=0.d.f(x)𝑓(𝑥) is differentiable at all real numbers x𝑥.

Identify the statement that is FALSE:If a function 𝑓 is continuous on (𝑎,𝑏), then 𝑓 always has an absolute minimum on (𝑎,𝑏).If a function 𝑓 is continuous on [𝑎,𝑏], then 𝑓 always has an absolute minimum on [𝑎,𝑏].If a function 𝑓 is continuous on [𝑎,𝑏] and 𝑓 has no relative extrema in (𝑎,𝑏), then the absolute maximum of 𝑓 on [𝑎,𝑏] exists and occurs at either 𝑥=𝑎 or 𝑥=𝑏.If a function 𝑓 has an absolute minimum value on (𝑎,𝑏), then there is a critical point of 𝑓 in (𝑎,𝑏).

Find the value of the constant 𝑎 that will make this piecewise function continuous everywhere.𝑓(𝑥)=⎧⎩⎨⎪⎪𝑥+1𝑎1−𝑥𝑥<0𝑥=0𝑥>0

If ∂f∂x=0∂𝑓∂𝑥=0, the function f(x,y)𝑓(𝑥,𝑦) has no dependence on the variable x𝑥.Select one:TrueFalse