Question

Use a graph to estimate the critical numbers of \(f(x) = \left| {{x^3} - 3{x^2} + 2} \right|\) correct to one decimal place.

Short answer

The critical numbers of function \(f(x) = \left| {{x^3} - 3{x^2} + 2} \right|\) are \( - 0.8,\;0,\;1,\;2,\,2.7\).

Step-by-step solution

Given data

The function is \(f(x) = \left| {{x^3} - 3{x^2} + 2} \right|\).

Concept of power rule

If two functions\(g(x)\)and\(h(x)\)are differentiable, then the derivative of\(f(x) = h(x)g(x)\)is,\({f^\prime }(x) = h(x){g^\prime }(x) + g(x){h^\prime }(x)\).v

Obtain the critical numbers of function \(f(x) = \left| {{x^3} - 3{x^2} + 2} \right|\)

The graph of the given function is shown below.

From the above graph, it can be inferred that, the critical points are\(x = - 0.8,\;x = 0,\;x = 1,\;x = 2,\;x = 2.7\).

Therefore, the critical numbers of function \(f(x) = \left| {{x^3} - 3{x^2} + 2} \right|\) are \( - 0.8,\;0,\;1,\;2,\,2.7\).