Question

Question 14.
Please just find the condition for the existence of the derivative at x=1​

Answer 1

Ok, so for f(x) = ax² + 1

$\lim_{x \to \ 1} (ax^{2} + 1)$

a + 1 -------------> LHL

Now, for f(x) = x + a

$\lim_{x \to \ 1} (x + a)$

1 + a --------------> RHL

Since the LHL is equal to RHL, the function f(x) exists with a derivative at x = 1

The only condition for derivative at x = 1 is that x $\neq$ 0 since if x = 0, then the derivative will have LHL $\neq$ RHL.