Find all the critical points of the function f(x) = 5x^{26} + 12x^{5} - 60x^{4} + 56. Use the First and/or Second Derivative Test to determine whether

Nann 2021-01-08 Answered
Find all the critical points of the function f(x)=5x26 + 12x5  60x4 + 56. Use the First and/or Second Derivative Test to determine whether each critical point is a local maximum, a local minimum, or neither. You may use either test, or both, but you must show your use of the test(s). You do not need to identify any global extrema.
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Expert Answer

sweererlirumeX
Answered 2021-01-09 Author has 91 answers

Step 1
Consider the given function as f(x)=5x6 + 12x5  60x4 + 56.
Note that the domain of the polynomial function f(x)=5x6 + 12x5  60x4 + 56 is (, ).
To identify the critical points, we have to solve the equation f(x)=0 as follows.
f(x)=5x6 + 12x5  60x4 + 56
 f(x)=5 × 6x5 + 12 × 5x4  60 × 4x3
 f(x)=30x5 + 60x4  240x3
Therefore,
f(x)=0
 30x5 + 60x4  240x3=0
 30x3(x2 + 2x  8)=0
 x3=0, x2 + 2x  8=0
 x=0, (x + 4)(x  2)=0
 x=0, x= 4, x=2
Hence, the function f(x)=5x6 + 12x5  60x4 + 56 has critical points at
x= 4, x=0 and x=2.
To get the y coordinate of the critical points, evaluate the function value at x= 4, x=0 and x=2.
When x= 4, the value:
f(4)=5(4)6 + 12(4)5  60(4)4 + 56= 7112.
When x=0, the value:
f(0)=5(0)6 + 12(0)5  60(0)4 + 56=56
When x=2, the value:
f(2)=5(2)6 + 12(2)5  60(2)4 + 56= 200.
Thus, the critical points of the function f(x)=5x6 + 12x5  60x4 + 56 are (4, 7112), (0, 56) and(2, 200).
Step 2
To identify whether the each of the above critical point is a local maximum, a local minimum, or neither, we use the first derivative test.
Split the domain (, ) using the critical points:
x= 4, x=0 and x=2 as (, 4), (4, 0), (0, 2) and(2, ).
Check the sign of the first derivative

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