Question

Find values for c and d that make h(x) continuous. 2x if x < 1 h(x) = cx?+d if 1sxs 2 4x if x>2

a. c=d=0
b. c = 2 and d = 0
c. с c= 10 and d = -8
d. c = 8 and d = 10

Answer 1

To make the function h(x) continuous, we need to find values for c and d. By setting up a system of equations using the given conditions and solving for the variables, we find that c = 2 and d = -8.

Step-by-step explanation:

To make the function h(x) continuous, we need to find values for c and d. First, let's examine the given conditions:

1. For x < 1, h(x) = 2x
2. For 1 <= x <= 2, h(x) = cx^2 + d
3. For x > 2, h(x) = 4x

To make the function continuous at x = 1, we need to make sure that the limit as x approaches 1 from the left is equal to the function value at x = 1. This means we need to set 2(1) = c(1)^2 + d. Simplifying, we have 2 = c + d.

To make the function continuous at x = 2, we need to make sure that the limit as x approaches 2 from the right is equal to the function value at x = 2. This means we need to set 4(2) = c(2)^2 + d. Simplifying, we have 8 = 4c + d.Now, we have a system of two equations with two variables:To find values for c and d that make h(x) continuous, we need to make sure the function h(x) has no jumps or breaks at the points where the definition changes, which are at x = 1 and x = 2. The given piecewise function has three parts, and we want the value of h(x) to be the same when approaching x = 1 from the left and right, and similarly at x = 2.

For x = 1, h(x) switches from 2x to cx^2 + d. Therefore, we want 2(1) = c(1)^2 + d, which simplifies to 2 = c + d. At x = 2, h(x) switches from cx^2 + d to 4x. Hence, we require c(2)^2 + d = 4(2), simplifying to 4c + d = 8. By solving these two equations together, we can find the values for c and d that will make h(x) continuous.

Solving for c and d in the equations 2 = c + d and 4c + d = 8 will give us the values needed for continuity. Hence, the correct answer is c = 2 and d = 0, which matches option b and ensures that h(x) is continuous across the entire domain.

1. 2 = c + d

2. 8 = 4c + d

Solving this system of equations, we find c = 2 and d = -8. Therefore, the correct values for c and d that make h(x) continuous are c = 2 and d = -8.