If f(x) is a continuous function, find the constants a and b such that
f(x) = { -x -3 <= x <= -2
ax^2 + b -2<x<0
b x=0
To find the values of constants a and b for the function f(x), you need to consider the given conditions for different intervals of x.
Condition 1: -3 ≤ x ≤ -2
In this interval, f(x) is defined as -x. Since f(x) is continuous, the value of f(-2) should be equal to the value of f(-3). Therefore,
f(-2) = -(-2) = 2
f(-3) = -(-3) = 3
Condition 2: -2 < x < 0
In this interval, f(x) is defined as ax^2 + b. Since f(x) is continuous, the value of f(-2) should be equal to the value of f(0). Therefore,
f(-2) = a(-2)^2 + b = 4a + b
f(0) = a(0)^2 + b = b
To find the value of a and b, equate the two expressions obtained:
4a + b = b
Simplifying, we find that 4a = 0. Therefore, a = 0.
Condition 3: x = 0
When x = 0, f(x) is defined as b. Using this condition, we can determine the value of b:
f(0) = b = 0
Therefore, the constants a and b for the function f(x) are a = 0 and b = 0.