document.write( "Question 1142315: Calculus Question\r
\n" ); document.write( "\n" ); document.write( "Consider the function \"+f%28x%29=3x%5E5-5x%5E3%2B3+\".\r
\n" ); document.write( "\n" ); document.write( "Use the first derivative to determine where the function is increasing or decreasing.\r
\n" ); document.write( "\n" ); document.write( "Find the local maximum and minimum values. Show your work.\r
\n" ); document.write( "\n" ); document.write( "Determine the intervals where the function is concave up or concave down.\r
\n" ); document.write( "\n" ); document.write( "Provide a sketch of the graph below. Label intercepts and critical points. \n" ); document.write( "
Algebra.Com's Answer #763047 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "f(x) = 3x^5-5x^3+3

\n" ); document.write( "f'(x) = 15x^4-15x^2 = 15x^2(x^2-1) = 15(x^2)(x-1)(x+1)

\n" ); document.write( "f''(x) = 60x^3-30x = 30x(x^2-1) = 30(x)(x-1)(x+1)

\n" ); document.write( "The critical points for both first and second derivatives are -1, 0, and 1; the resulting intervals are (-infinity, -1), (-1,0), (0,1), and (1, infinity).

\n" ); document.write( "An analysis, using the factored forms of the first and second derivatives and test points in each interval:
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document.write( "   interval               f'(x)                           f''(x)\r\n" );
document.write( " ---------------------------------------------------------------------------------\r\n" );
document.write( "  (-infinity, -1)  (+)(+)(-)(-) = +  increasing    (+)(-)(-)(-) = -  concave down\r\n" );
document.write( "  (-1,0)           (+)(+)(+)(-) = -  decreasing    (+)(-)(+)(-) = +  concave up\r\n" );
document.write( "  (0,1 )           (+)(+)(+)(-) = -  decreasing    (+)(+)(+)(-) = -  concave down\r\n" );
document.write( "  (1,infinity)     (+)(+)(+)(+) = -  increasing    (+)(+)(+)(+) = +  concave up

\n" ); document.write( "A graph confirming the preceding analysis:

\n" ); document.write( "\"graph%28400%2C400%2C-2%2C2%2C-10%2C10%2C3x%5E5-5x%5E3%2B3%29\"

\n" ); document.write( "NOTE: While most resources use test points in each interval to determine increasing/decreasing and concave up/concave down, there is an easier and faster way to make those determinations.

\n" ); document.write( "Consider \"walking\" along the number line and seeing what happens to the signs of the factors of the derivatives each time you pass one of the critical points.

\n" ); document.write( "First derivative test for increasing/decreasing:

\n" ); document.write( "For large negative values of x, f'(x) = 15(x^2)(x-1)(x+1) is positive; the function is increasing.
\n" ); document.write( "When we pass x=-1, one factor in f'(x) changes sign, so the sign of f'(x) changes; the function is now decreasing.
\n" ); document.write( "When we pass x=0, TWO factors change sign, resulting in no change to the sign of f'(x); the function is still decreasing.
\n" ); document.write( "When we pass x=1, one factor changes sign, so the sign of f'(x) changes; the function is now increasing.

\n" ); document.write( "Summary: increasing on (-infinity,-1) and (1,infinity); decreasing on (-1,0) and (0,1).

\n" ); document.write( "Second derivative test for concave up/concave down:

\n" ); document.write( "For large negative values of x, f''(x) = 30(x)(x-1)(x+1) is negative; the function is concave down.
\n" ); document.write( "When we pass each of the three critical points, one factor in f''(x) changes sign, so the concavity changes at each critical point.

\n" ); document.write( "Summary: concave down on (-infinity,-1) and (0,1); concave up on (-1,0) and (1,infinity).
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