Consider the function f(x)=x^4-32x^2+11, -3 <= x <= 9. This function has an absolute minimum value equal to ? and an absolute maximum value equal to ?

sjeikdom0 2020-12-21 Answered
Consider the function f(x)=x432x2+11,3x9. This function has an absolute minimum value equal to ? and an absolute maximum value equal to ?
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Expert Answer

Bertha Stark
Answered 2020-12-22 Author has 96 answers
Step 1
Consider the given function,
f(x)=x432x2+11,3x9
differentiate the function with respect to x,
f(x)=ddx(x432x2+11)
=ddx(x4)ddx(32x2)+ddx(11)
=4x364x
for the maxima and minima, f′(x)=0,
4x364x=0
4x(x216)=0
x=0,-4,4
the values of x lie in the interval 3x9 is,
x=0,4
Step 2
Now, use the second derivative test so differentiate again,
f(x)ddx(4x364x)
=12x264
at x = 0,
f(x)=12(0)264
=-64
f''(x)<0
so, the function has the absolute maximum at x=0,
now, for the absolute maximum value substitute x=0 in the function,
f(0)=((0)4)32(0)2+11
=0-0+11
=11
the absolute maximum value of the function is 11.
Step 3
at x=4,
f(x)=12(4)264
=12×1664
=128
f''(x)>0
so, the function has the absolute minimum at x=4,
now, for the absolute minimum value substitute x=4 in the function,
f(4)=(4)432(4)2+11
=256-512+11
=-245
the absolute minimum value of the function is −245.
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