Respuesta :
Take the first derivative to find critical points:
[tex]f(x)=x^5-10x^3+9x\implies f'(x)=5x^4-30x^2+9=0[/tex]
[tex]\implies x^2=3\pm\dfrac6{\sqrt5}\implies x=\pm\sqrt{3\pm\dfrac6{\sqrt5}}[/tex]
or approximately (from least to greatest) -2.4, -0.56, 0.56, 2.4.
We have second derivative
[tex]f''(x)=20x^3-60x[/tex]
and at each of the critical points, we have
[tex]f''(-2.4)\approx-128<0[/tex]
[tex]f''(-0.56)\approx30>0[/tex]
[tex]f''(0.56)\approx-30<0[/tex]
[tex]f''(2.4)\approx128>0[/tex]
The signs of the second derivative at each point indicates a local minima at [tex]x\approx-2.4[/tex] and [tex]x\approx0.56[/tex], and local maxima at [tex]x\approx-0.56[/tex] and [tex]x\approx2.4[/tex]. At these extrema, we have
[tex]f(-2.4)\approx37.014[/tex]
[tex]f(-0.56)\approx-3.34[/tex]
[tex]f(0.56)\approx3.34[/tex]
[tex]f(2.4)\approx-37.014[/tex]
and at the endpoints of the interval, we have
[tex]f(-3)=f(3)=0[/tex]
So the answer is A.
The approximate values of the minimum and maximum points
maximum point: (–2.4, 37.014) and minimum point: (2.4, –37.014).
Approximate values
Given function
[tex]$\mathrm{f}(\mathrm{x})=x^{5}-10 x^{3}+9 x$[/tex]on the interval [-3,3]
A. Maximum point (-2.4,37.014)
Minimum point (2.4,-37.014)
[tex]$f(x)=(-2.4)^{5}-10(-2.4)^{3}+9(-2.4)$[/tex]
[tex]$f(x)=-79.62624+138.24-21.6$[/tex]
[tex]$f(x)=37.014$[/tex]
Put x=2.4 then we get
[tex]&f(x)=(2.4)^{5}-10(2.4)^{3}+9(2.4) \\[/tex]
[tex]&f(x)=79.62624-138.24+21.6 \\[/tex]
[tex]&f(x)=37.014[/tex]
B. Maximum point (2.4,-37.014)
Minimum point (-2.4,37.014)
Put x=2.4. Then we get
f(x)=-37.014
Put x=-2.4 then we get
f(x)=37.014
C. Maximum point $(-1.4,33.014)$
Minimum point $(1.4,-33.014)$
Put x=-1.4 then we get
[tex]&f(x)=(-1.4)^{5}-10(-1.4)^{3}+9(1.4) \\[/tex]
[tex]&f(x)=-38.94[/tex]
Put x =1.4 then we get
f(x) = 38.94
D. Maximum point (-3,30)
Minimum point (3,-30)
f(x) = 38.94
Put [tex]$x=-3$[/tex] then we get
[tex]$f(x)=-243+270-27=0$[/tex]
Hence, from the option [tex]$A, B, C$[/tex] and [tex]D[/tex] we can see only option [tex]$A$[/tex] is the right answer. The approximate values of the minimum point [tex]$(2.4,-37.014)$[/tex]and maximum point [tex]$(-2.4,37.014)$[/tex] of the function [tex]$\mathrm{f}(\mathrm{x})=x^{5}-10 x^{3}+9 x$[/tex] on [tex]$[-3,3]$[/tex].
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