Question

Let f(x) = 4x^3-5x^2Then f(x) has a local minimum at x= ____a local maximum at x= ____and inflection point at x= ____ write inflection points (if any) in numerical order smallest first

Answer 1

Given:

`f\mleft(x\mright)=4x^3-5x^2`

Find-: Local minimum and local maximum and inflection point.

Sol:

Derivative of function.

`\begin{gathered} f\mleft(x\mright)=4x^3-5x^2 \\ f^(\prime)\left(x\right)=12x^2-10x \\ f^(\prime)\left(x\right)=2x\left(6x-5\right) \end{gathered}`

The critical point is:

`\begin{gathered} f^(\prime)\left(x\right)=0 \\ 2x\left(6x-5\right)=0 \\ 2x=0;6x-5=0 \\ x=0;x=(5)/(6) \end{gathered}`

Local minima is:

`\left(x,f\lparen x\right))=\lparen(5)/(6),-1.157)`

Local minima at x=5/6

Local maxima at x=0

Inflection point.

`\begin{gathered} f=4x^3-5x^2 \\ \text{ Inflection point} \\ x=(5)/(12) \end{gathered}`