inspect the graph of the function to determine whether it is concave up, concave down or neither, on the…

inspect the graph of the function to determine whether it is concave up, concave down or neither, on the given interval.\na cube function, ( m(x)=-5 x^{3} ), on ( (0, infty) )\non the interval ( (0, infty) ), the function ( m(x)=-5 x^{3} ) is

inspect the graph of the function to determine whether it is concave up, concave down or neither, on the given interval.\na cube function, ( m(x)=-5 x^{3} ), on ( (0, infty) )\non the interval ( (0, infty) ), the function ( m(x)=-5 x^{3} ) is

Answer

Explanation:

Step1: Find the first derivative

The derivative of (m(x)=-5x^{3}) is (m^{\prime}(x)=-15x^{2}) (using the power rule ((x^{n})^\prime = nx^{n - 1})).

Step2: Find the second derivative

Differentiate (m^{\prime}(x)=-15x^{2}) with respect to (x). Using the power rule again, (m^{\prime\prime}(x)=-30x).

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

For (x\in(0,\infty)), if (x>0), then (m^{\prime\prime}(x)=-30x<0).

Answer:

The function (m(x)=-5x^{3}) is concave down on the interval ((0,\infty)).