Let f (x) = {x ^ 2, X ≤ 1; ax + B, x > 1} to make f (x) continuous and differentiable at x = 1, what values should a and B take?

Let f (x) = {x ^ 2, X ≤ 1; ax + B, x > 1} to make f (x) continuous and differentiable at x = 1, what values should a and B take?

F (1) = 1 linf (x) x → 1 + = a + BX ≤ 1 f '(x) = 2x limf' (x) x → 1 - = 2x > 1 f '(x) = a limf' (x) x → 1 + = a is continuous at x = 1, f (1) = linf (x) x → 1 + 1 = a + B. (1) is continuous and differentiable at x = 1, limf '(x) x → 1 - = limf' (x) x → 1 + 2 = A. (2) solution (1) (2) a = 2, B = - 1 linf (x) x → 1 + denotes

Let f (x) = x ^ 2, X1 be differentiable at x = 1, and find the value of a and B

Can be continuous
f(1)=1^2=1
Then x tends to 1 + and ax + B is 1
So a + B = 1
The left and right derivatives can be equal
(x^2)'=2x
So the left derivative is 2
(ax+b)'=a
Then the right derivative = a = 2
therefore
a=2,b=1-a=-1

Derivative function f (x) = ax ^ 3-ax ^ 2 + [f '(1) / 2-1]
Given f (x) = ax ^ 3-ax ^ 2 + [f '(1) / 2-1]
Let a denote f '(1)
It should be f (x) = ax ^ 3-ax ^ 2 + [f '(1) / 2-1] X

f(x)=ax^3-ax^2+[f‘(1)/2-1]x
So f '(x) = 3ax ^ 2-2ax + F' (1) / 2-1
f'(1)=3a-2a+f'(1)/2-1
f'(1)=2a-2