Question

Select the correct answer. What are the approximate values of the minimum and maximum points of f(x) = x5 − 10x3 + 9x on [-3,3]? A. maximum point: (–2.4, 37.014) and minimum point: (2.4, –37.014) B. maximum point: (2.4, –37.014) and minimum point: (–2.4, 37.014) C. maximum point: (–1.4, 33.014) and minimum point: (1.4, –33.014) D. maximum point: (–3, 30) and minimum point: (3, –30)

Answer 1

Take the first derivative to find critical points:

`f(x)=x^5-10x^3+9x\implies f'(x)=5x^4-30x^2+9=0`

`\implies x^2=3\pm\frac6{\sqrt5}\implies x=\pm\sqrt{3\pm\frac6{\sqrt5}}`

or approximately (from least to greatest) -2.4, -0.56, 0.56, 2.4.

We have second derivative

`f''(x)=20x^3-60x`

and at each of the critical points, we have

`f''(-2.4)\approx-128<0`

`f''(-0.56)\approx30>0`

`f''(0.56)\approx-30<0`

`f''(2.4)\approx128>0`

The signs of the second derivative at each point indicates a local minima at
`x\approx-2.4` and
`x\approx0.56`, and local maxima at
`x\approx-0.56` and
`x\approx2.4`. At these extrema, we have

`f(-2.4)\approx37.014`

`f(-0.56)\approx-3.34`

`f(0.56)\approx3.34`

`f(2.4)\approx-37.014`

and at the endpoints of the interval, we have

`f(-3)=f(3)=0`

So the answer is A.