Let f(x)=x3−6x2−15x+23 a) Perform a full sign analysis on f(x): find its zeroes, and determine its sign on each interval. b) Use the information in (2) to determine intervals of concave up / concave down for f. c) Find the coordinates (x, y) of the inflection point(s) of f. d) Find the absolute maximum and minimum values of f on the interval [0, 7]. Justify your answer with appropriate calculus work.

Question

Let f(x)=x3−6x2−15x+23  a) Perform a full sign analysis on f(x): find its zeroes, and determine its sign on each interval.   b) Use the information in (2) to determine intervals of concave up / concave down for f.   c) Find the coordinates (x, y) of the inflection point(s) of f.   d) Find the absolute maximum and minimum values of f on the interval [0, 7]. Justify your answer with appropriate calculus work.

Ruth Phillips · Accepted Answer

Step 1Given:f(x)=x3−6x2−15x+23f′(x)=3x2−12x−15a) f″(x)=6x−12=0⇒x=2b) Concave up on interval (2, ∞)Concave dow on interval (−∞, 2)c) Cooedinate of inplection point (x, y)F(2)=23−6×22−15×2+23=8−24−30+23=−23(x, y)=(2, −23)d) F′(x)=0⇒3x2−12x−15=0⇒x2−4x−5=0x2+x−5x−5=0x(x+1)−5(x+1)=0(x+1)(x−5)=0⇒x=−1 and x=5F(−1)=(−1)3−6×(−1)2−15×−1+23=−1−6+15+23=31F(5)=53−6×52−15×5+23=125−150−75+23=−77F(7)=73−6×72−15×7+23=343−294−105+23=−33F(0)=03−6×02−15×0+23=0+0+0+23=23Absolute maxima at x=−1 and maximum volue is 31Absolute minima at x=5 and minimum volue is −77

Let f(x)=x3−6x2−15x+23 a) Perform a full sign analysis on f(x): find its zeroes, and determine...

Answered question

Answer & Explanation