b. Investigate continuity of the function f(x) at x = 1, Where x² +1 , xs1 , x>1 f«) = {*2x

College Algebra (MindTap Course List)
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Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
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hey, can you help me to solve these calculus questions (blue marked only)? I need every step to be shown of the answers. If can then please let me know TIA

1. a. Write down the conditions for a function f(x) to be continuous at a point x = c.
b. Investigate continuity of the function f(x) at x = 1, Where
f(x) = { x²+1 , x s1
2x
, x>1
a. Verify the Rolle's Theorem for the function f(x) = x² in (-1, 1).
b. If y = tanx¸ then show that
(1+ x?)yn+2 + 2(n + 1)xyn+1 + (n² +n)yn = 0
3. a. Find f(x), if it exists. Where f(x) =
S2x +1 , x s1
3-x , x> 1
lim
b. Find yn, where y =
6x2-5x+1'
a. Investigate differentiability of the function f(x) at x = 0, Where
3+ 2x ,
3
<xs0
f(x) = •
3— 2х,
3
0<x <;
b. Verify the Mean Value Theorem for the function f(x) = x³ +1 in (1, 2).
2.
Transcribed Image Text:1. a. Write down the conditions for a function f(x) to be continuous at a point x = c. b. Investigate continuity of the function f(x) at x = 1, Where f(x) = { x²+1 , x s1 2x , x>1 a. Verify the Rolle's Theorem for the function f(x) = x² in (-1, 1). b. If y = tanx¸ then show that (1+ x?)yn+2 + 2(n + 1)xyn+1 + (n² +n)yn = 0 3. a. Find f(x), if it exists. Where f(x) = S2x +1 , x s1 3-x , x> 1 lim b. Find yn, where y = 6x2-5x+1' a. Investigate differentiability of the function f(x) at x = 0, Where 3+ 2x , 3 <xs0 f(x) = • 3— 2х, 3 0<x <; b. Verify the Mean Value Theorem for the function f(x) = x³ +1 in (1, 2). 2.
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