(1 point) Find a and b so that the function f(x)= ) 5x^3-6x^2+8,&xlt -2 ax+b,&xgeqslant -2 is both continuous and differentiable. a= b=

Question

Answer 1

To ensure that the function

is both continuous and differentiable at

, we need to make sure that the left-hand limit and the right-hand limit are equal at

, and that the derivatives from both sides are also equal at that point.First, let's find the left-hand limit as

approaches

:

Next, let's find the right-hand limit as

approaches

:

For the function to be continuous at

, the left-hand limit and the right-hand limit must be equal:

Now, let's find the derivative of the left-hand side of the function:

Next, let's find the derivative of the right-hand side of the function:

For the function to be differentiable at

, the derivatives from both sides must be equal:

Now, substitute

into the equation

to find

:

Therefore, the values of

and

that make the function

both continuous and differentiable are:

Soru