Question

inspect the graph of the function to determine whether it is concave up, concave down or neither, on the given interval.
a cube function, ( m(x)=4x^{3} ), on ( (0,infty) )
on the interval ( (0,infty) ), the function ( m(x)=4x^{3} ) is

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(m(x)=4x^{3}\), \(m^\prime(x)=4\times3x^{2}=12x^{2}\).

Step2: Find the second derivative

Differentiate \(m^\prime(x)=12x^{2}\) again. Using the power rule, \(m^{\prime\prime}(x)=12\times2x = 24x\).

Step3: Analyze the sign of the second derivative on \((0,\infty)\)

For \(x\in(0,\infty)\), \(m^{\prime\prime}(x)=24x>0\).

Answer:

concave up