Yes, the production function is concave. To show this, we can take the second derivative of the production function:
d2Q/dX2 = -1/2X-3/2
Since this is always negative, the production function is concave.
a) is the production function concave? Show your working
d2Q/dX2 = -1/2X-3/2
Since this is always negative, the production function is concave.
First, let's find the first derivative of the production function:
dQ/dX = (1/2) * X^(-1/2)
Next, we can find the second derivative by taking the derivative of the first derivative:
d^2Q/dX^2 = (-1/4) * X^(-3/2)
Now, we need to check if the second derivative is negative for all values of X. Since the second derivative has a negative constant factor (-1/4), and X raised to a negative power, we can conclude that the second derivative is indeed negative for all X.
Therefore, the production function Q = X^(1/2) is concave.
First, let's find the first derivative of the production function, Q, with respect to X:
dQ/dX = (1/2)X^(-1/2)
Now, let's find the second derivative of Q with respect to X:
d^2Q/dX^2 = (d/dX) ((1/2)X^(-1/2))
Using the power rule for differentiation, we can find the second derivative:
d^2Q/dX^2 = (1/2) * (-1/2) * X^(-3/2)
Simplifying the expression for the second derivative:
d^2Q/dX^2 = -1/4 * X^(-3/2)
Since the second derivative is negative, we can conclude that the production function is concave.