Absolute maxima and minima Determine the location and value of

Zerrilloh6

Zerrilloh6

Answered question

2021-12-14

Absolute maxima and minima Determine the location and value of the absolute extreme value of f on the given interval, if they exist.
f(x)=2x315x2+24 on [0,5]

Answer & Explanation

Nadine Salcido

Nadine Salcido

Beginner2021-12-15Added 34 answers

Step 1: Given
f(x)=2x315x2+24
And the interval is [0, 5]
Step 2:
f(x)=2x315x2+24x
Differentiate with respect to x,
f(x)=6x230x+24
f'(x) is a polynomial, so every critical point of f(x) is zero of f'(x).
f(x)=2x315x2+24x
At x=1
f(1)=2(1)315(1)2+24(1)=215+24=11
At x=4
f(x)=2(4)315(4)2+24(4)=16
Step 3:
To find zeros of f'(x)
f'(x)=0
6x230x+24=0
x25x+4=0
x24xx+4=0
x(x-4)-1(x-4)=0
(x-4)(x-1)=0
x=1,4
Hence the zeros of f'(x) are x=1, 4
Step 4:
1,4[0,5]
Evaluate f(x) at each critical point.
At x=1
f(1)=2(1)315(1)2+24(1)=11
At x=4
f(4)=2(4)315(4)2+24(4)=16
The minimum is -16 occurs at x=4.
The maximum is 11 occurs at x=1.

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