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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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.

sweererlirumeX · Accepted Answer

Step 1Consider 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 − 240x3Therefore,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=2Hence, the function f(x)=5x6 + 12x5 − 60x4 + 56 has critical points atx= −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=56When 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 2To 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

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

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