Piecewise function problem.
Let f(x)={ax^2+1/3, x is greater than or equal to 1; bx-10/3, x<1. If the function is differentiable, find the sum of a+b.
f(x) =
ax^2 + 1/3 for x >= 1
bx - 10/3 for x < 1
so, we need f(1) to be consistent. That is,
a + 1/3 = b - 10/3
we also need f'(1) to be consistent:
2a = b
so, a + 1/3 = 2a - 10/3
a = 11/3
b = 22/3
a+b = 11
To determine the sum of a and b, we need to find the values of a and b that make the function f(x) differentiable at x = 1.
For a function to be differentiable at a point, it needs to have a derivative that exists at that point. In this case, we need to ensure that the derivatives of the two pieces of the function match at x = 1.
Let's find the derivatives of both pieces of the function.
For the first piece, f(x) = ax^2 + 1/3:
To find the derivative, we differentiate the function with respect to x:
f'(x) = 2ax.
For the second piece, f(x) = bx - 10/3:
Similarly, we find the derivative:
f'(x) = b.
To ensure that the function is differentiable at x = 1, the derivatives of the two pieces must be equal at that point:
f'(1) = f'(1).
Set the derivatives equal to each other and solve for a and b:
2a(1) = b.
2a = b.
Since both derivatives are equal, we have the equation 2a = b.
Now, let's combine this equation with the given function values for f(x) at x = 1.
For x ≥ 1: f(x) = ax^2 + 1/3.
Since we know x = 1, substitute it into the equation:
f(1) = a(1)^2 + 1/3.
f(1) = a + 1/3.
For x < 1: f(x) = bx - 10/3.
Since we know x = 1, substitute it into the equation:
f(1) = b(1) - 10/3.
f(1) = b - 10/3.
Since both pieces of the function should be equal at x = 1, set the two equations equal to each other:
a + 1/3 = b - 10/3.
Now, substitute the relationship between a and b we found earlier:
2a = b.
a + 1/3 = 2a - 10/3.
Simplify and solve for a:
1/3 + 10/3 = 2a - a.
11/3 = a.
Given that a = 11/3, substitute it back into the relationship between a and b:
2(11/3) = b.
22/3 = b.
Thus, the values of a and b that make the function differentiable at x = 1 are a = 11/3 and b = 22/3.
To find the sum of a and b:
a + b = 11/3 + 22/3 = 33/3 = 11.
Therefore, the sum of a and b is 11.