Question

consider the function ( f(x)=12 x^{5}+60 x^{4}-240 x^{3}+5 ).
( f(x) ) has inflection points at (reading from left to right) ( x=d, e ), and ( f )
where ( d ) is
and ( e ) is
and ( f ) is
for each of the following intervals, tell whether ( f(x) ) is concave up or concave down.
( (-infty, d) ): select an answer
( (d, e) ): select an answer
( (e, f) ): select an answer
( (f, infty) ): select an answer
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Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(y = f(x)=12x^{5}+60x^{4}-240x^{3}+5\), we have \(f^\prime(x)=60x^{4}+240x^{3}-720x^{2}\)

Step2: Find the second derivative

Differentiate \(f^\prime(x)\) again. \(f^{\prime\prime}(x)=240x^{3}+720x^{2}-1440x\)
Factor out \(240x\): \(f^{\prime\prime}(x)=240x(x^{2}+3x - 6)\)
Set \(f^{\prime\prime}(x)=0\). We solve \(240x(x^{2}+3x - 6)=0\)
The solutions of \(x^{2}+3x - 6=0\) are \(x=\frac{-3\pm\sqrt{9+24}}{2}=\frac{-3\pm\sqrt{33}}{2}\approx\frac{-3\pm5.74}{2}\)
The roots of \(f^{\prime\prime}(x) = 0\) are \(x = 0\), \(x=\frac{-3-\sqrt{33}}{2}\approx - 4.37\), \(x=\frac{-3 + \sqrt{33}}{2}\approx1.37\)

Step3: Test the intervals

• For the interval \((-\infty,\frac{-3-\sqrt{33}}{2})\), let \(x=-5\). Then \(f^{\prime\prime}(-5)=240\times(-5)\times((-5)^{2}+3\times(-5)-6)=240\times(-5)\times(25 - 15 - 6)=240\times(-5)\times4=-4800<0\), so \(f(x)\) is concave down.
• For the interval \((\frac{-3-\sqrt{33}}{2},0)\), let \(x=-1\). Then \(f^{\prime\prime}(-1)=240\times(-1)\times((-1)^{2}+3\times(-1)-6)=240\times(-1)\times(1 - 3 - 6)=240\times(-1)\times(-8) = 1920>0\), so \(f(x)\) is concave up.
• For the interval \((0,\frac{-3+\sqrt{33}}{2})\), let \(x = 1\). Then \(f^{\prime\prime}(1)=240\times1\times(1^{2}+3\times1 - 6)=240\times1\times(1 + 3 - 6)=240\times(-2)=-480<0\), so \(f(x)\) is concave down.
• For the interval \((\frac{-3+\sqrt{33}}{2},\infty)\), let \(x = 2\). Then \(f^{\prime\prime}(2)=240\times2\times(2^{2}+3\times2 - 6)=240\times2\times(4 + 6 - 6)=960>0\), so \(f(x)\) is concave up.

Answer:

\(D=\frac{-3-\sqrt{33}}{2}\), \(E = 0\), \(F=\frac{-3+\sqrt{33}}{2}\)
\((-\infty,D)\): concave down
\((D,E)\): concave up
\((E,F)\): concave down
\((F,\infty)\): concave up