for each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where…

for each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. using this information, sketch the graph of the function.\n1) $y = -dfrac{x^3}{3} + x^2$

for each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. using this information, sketch the graph of the function.\n1) $y = -dfrac{x^3}{3} + x^2$

Answer

Explanation:

Step1: Find x and y intercepts

  • x-intercepts: Set ( y = 0 ), so ( -\frac{x^3}{3}+x^2 = 0 ). Factor out ( x^2 ): ( x^2\left(-\frac{x}{3}+1\right)=0 ). Solutions are ( x = 0 ) and ( -\frac{x}{3}+1 = 0\Rightarrow x = 3 ).
  • y-intercept: Set ( x = 0 ), then ( y = -\frac{0^3}{3}+0^2 = 0 ).

Step2: Find critical points (derivative = 0)

  • First derivative: ( y'=-\frac{3x^2}{3}+2x=-x^2 + 2x ).
  • Set ( y' = 0 ): ( -x^2 + 2x = 0 ). Factor: ( x(-x + 2)=0 ). Solutions: ( x = 0 ) and ( x = 2 ).

Step3: Intervals of increase/decrease

  • Test intervals: ( (-\infty, 0) ), ( (0, 2) ), ( (2, \infty) ).
  • For ( (-\infty, 0) ), pick ( x=-1 ): ( y'=-(-1)^2+2(-1)=-1 - 2=-3<0 ) (decreasing).
  • For ( (0, 2) ), pick ( x = 1 ): ( y'=-1^2+2(1)=1>0 ) (increasing).
  • For ( (2, \infty) ), pick ( x = 3 ): ( y'=-3^2+2(3)=-9 + 6=-3<0 ) (decreasing).

Step4: Relative minima/maxima

  • At ( x = 0 ): left decreasing, right increasing? Wait, left of ( 0 ) is decreasing, right (0,2) is increasing? No, wait: left of ( 0 ) (e.g., ( x=-1 )): ( y' < 0 ) (decreasing), right of ( 0 ) (e.g., ( x = 1 )): ( y' > 0 ) (increasing)? Wait, no: at ( x = 0 ), left is decreasing, right is increasing? Wait, no: when ( x ) moves from left of ( 0 ) to right of ( 0 ), ( y' ) goes from negative to positive? Wait, no: at ( x = 0 ), left (( x=-1 )): ( y'=-3<0 ), right (( x = 1 )): ( y'=1>0 ). Wait, but at ( x = 0 ), the function changes from decreasing to increasing? Wait, no, wait the derivative at ( x = 0 ): ( y'(-1)=-3 ), ( y'(1)=1 ). So at ( x = 0 ), the function has a local minimum? Wait, no, wait when ( x ) is less than ( 0 ), ( y' < 0 ) (decreasing), between ( 0 ) and ( 2 ), ( y' > 0 ) (increasing), so at ( x = 0 ), the function changes from decreasing to increasing: local minimum? Wait, but at ( x = 2 ): left (( x = 1 )) ( y' > 0 ) (increasing), right (( x = 3 )) ( y' < 0 ) (decreasing), so at ( x = 2 ), local maximum.
  • Calculate ( y ) at critical points:
    • At ( x = 0 ): ( y = 0 ) (local minimum? Wait, no, wait when ( x < 0 ), the function is decreasing, at ( x = 0 ), then increasing. Wait, but let's check ( y ) values. At ( x = 0 ), ( y = 0 ); at ( x = 2 ), ( y=-\frac{8}{3}+4=\frac{4}{3}\approx1.333 ).

Step5: Second derivative (concavity, inflection points)

  • Second derivative: ( y''=-2x + 2 ).
  • Set ( y'' = 0 ): ( -2x + 2 = 0\Rightarrow x = 1 ).

Step6: Intervals of concavity

  • Test intervals: ( (-\infty, 1) ), ( (1, \infty) ).
  • For ( (-\infty, 1) ), pick ( x = 0 ): ( y''=-0 + 2 = 2>0 ) (concave up).
  • For ( (1, \infty) ), pick ( x = 2 ): ( y''=-4 + 2=-2<0 ) (concave down).

Step7: Inflection point

  • At ( x = 1 ), ( y=-\frac{1}{3}+1=\frac{2}{3}\approx0.666 ). So inflection point is ( (1, \frac{2}{3}) ).

Summary:

  • x-intercepts: ( x = 0, 3 ); y-intercept: ( y = 0 ).
  • Critical points (x-coordinates): ( x = 0, 2 ).
  • Intervals of increase: ( (0, 2) ); decrease: ( (-\infty, 0) ), ( (2, \infty) ).
  • Relative minima: ( (0, 0) ); relative maxima: ( (2, \frac{4}{3}) ).
  • Inflection point (x-coordinate): ( x = 1 ).
  • Concave up: ( (-\infty, 1) ); concave down: ( (1, \infty) ).

To sketch the graph:

  • Plot intercepts ( (0,0) ), ( (3,0) ).
  • Plot critical points ( (0,0) ) (local min), ( (2, \frac{4}{3}) ) (local max).
  • Plot inflection point ( (1, \frac{2}{3}) ).
  • Use concavity and increase/decrease intervals to draw the curve: decreasing on ( (-\infty, 0) ), increasing on ( (0, 2) ), decreasing on ( (2, \infty) ); concave up on ( (-\infty, 1) ), concave down on ( (1, \infty) ).