In each item, do the following analytically: (a) find the relative extrema of f. (b) determine the values of x at which the relative extrema occur. (c) determine the intervals on which f is increasing. (d) determine the intervals on which f is decreasing. f(x)=x3−3x2−9x

Question

In each item, do the following analytically:   (a) find the relative extrema of f.  (b) determine the values of x at which the relative extrema occur.  (c) determine the intervals on which f is increasing.  (d) determine the intervals on which f is decreasing.  f(x)=x3−3x2−9x

StrycharzT · Accepted Answer

(a) Find the domain of the funtion f(x)=x3−3x2−9x, x∈R Finde the derivative f′(x)=3x2−6x−9 Find the domain of derivative f′(x)=3x2−6x−9, x∈R Substitute f&#039;(x)=0 0=3x2−6x−9 Solve equation x=−1x=3Determine th intervals ⟨−∞,−1⟩,⟨−1,3⟩⟨−1,3⟩,⟨3,+∞⟩ Choose the points x1=−2x2=0x3=0x4=0 Calculate the values of the derivatives f′(−2)=15f′(0)=−9f′(0)=−9f′(4)=15 The relative maximum at x=-1 The relative minimum is at x=3 f′(x)=3x2−6x−9, x=−1f′(x)=3x2−6x−9, x=3 Calculate the value f(−1)=5f(3)=-27 The relative maximum of the function is 5 at x=-1 The relative minimum of the function is -27 at x=3 (b) To find x-intercept, substitute f(x)=0 0=x3−3x2−9x Swap the sides x3−3x2−9x=0 Factor the expression x×(x2−3x−9)=0 Separate onto possible cases x=0x2−3x−9=0 Solve the equation x=0x=3+352x=3−352 There is 3 solutions (c) and (d) Let&#039;s plot the polynomial to determine the intervals of increasing and decreasing. Increase by: (−∞,−1),(3,∞) Decreases by: (−1,3)

In each item, do the following analytically:

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