Find the absolute maximum and absolute minimum value of f on the given interval. f(x) = 9+ 81x - 3x^3, [0,4]

EunoR

EunoR

Answered question

2021-02-03

Find the absolute maximum and absolute minimum value of f on the given interval.
f(x)=9+81x3x3,[0,4]

Answer & Explanation

mhalmantus

mhalmantus

Skilled2021-02-04Added 105 answers

Step 1
Consider the function
f(x)=9+81x3x3, on[0,4]
Step 2
Because the given function is a polynomial, it is continuous everywhere and thus on the given interval.
Critical points: Critical points of the function are the points where the derivative of the function either zero or does not exist.
Compute the derivative
f(x)=9+81x3x3
f(x)=819x2
Next to find the critical points.
f(x)=0
819x2=0
x=±3
Here only critical point x=3 lies in the interval [0,4]
Step 3
Then evaluate the function value at the critical points and end points.
f(x)=9+81x3x3
at x=0
f(0)=9
at x=3
f(3)=9+813333
f(3)=171
at x=4
f(4)=9+814343
f(4)=141
Absolute maximum value is the largest function value and the absolute minimum value is the smallest function value.
Absolute Maximum value of f(x)=171 at x=3
Absolute Minimum value of f(x)=9 at x=3.
Step 4
Absolute Maximum value =171
Absolute Minimum value =9

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