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).

f(c) is the maximum value of f on [a, b] if 𝑓(𝑐)≤𝑓(𝑥)

Consider the function:𝑓(𝑥)=𝑥3−3𝑥22−36𝑥Find the relative maximum point on this function.

Find the open intervals on which the function 𝑓 is increasing or decreasing, and find the 𝑥-values of all relative extrema (turning points).𝑓(𝑥)=18⋅𝑥−𝑥3

Select the correct answer.Consider function f.𝑓⁡(𝑥)={2𝑥,𝑥<0-𝑥2−4⁢𝑥+1,0<𝑥<212⁢𝑥+3,𝑥>2Which statement is true about function f? A. The function is continuous. B. The function is increasing over its entire domain. C. The domain is all real numbers. D. As x approaches positive infinity, 𝑓⁡(𝑥) approaches positive infinity.

𝑓(𝑥)={𝑥2−2𝑎𝑥+𝑎2𝑥−𝑎5,𝑖𝑓𝑥=𝑎,𝑖𝑓𝑥≠𝑎f(x)={ x−ax 2 −2ax+a 2 ​ 5,ifx=a​ ,ifx≠aWhich of the following are true about f?I. lim⁡𝑥→𝑎 𝑓(𝑥)  exists.II. 𝑓(𝑎)  exists.III.𝑓(𝑥)  is continuous at x = a​ I.  x→alim​  f(x)  exists.II. f(a)  exists.III.f(x)  is continuous at x = a​