Analyzing Graphs with the Second Derivative
Learning Objectives
- Intervals of increasing/decreasing
- Local extrema
- Applications
Exercises
1.
\[
\begin{align*}
f''(x) &= \dfrac{(10x^3-27x^2+15)^2(-160x^3+180x-42)}{(10x^3-27x^2+15)^4} \\
&- \dfrac{(-40x^4+90x^2-42x)(2(10x^3-27x^2+15)(30x^2-54x))}{(10x^3-27x^2+15)^4}\end{align*}\] Note: This answer is typically written as one fraction (see the video), but that fraction is too long for some display sizes on our website. To solve this, we rewrote it as the difference of two fractions and used two lines. Both functions are equal.
\begin{align*}
f''(x) &= \dfrac{(10x^3-27x^2+15)^2(-160x^3+180x-42)}{(10x^3-27x^2+15)^4} \\
&- \dfrac{(-40x^4+90x^2-42x)(2(10x^3-27x^2+15)(30x^2-54x))}{(10x^3-27x^2+15)^4}\end{align*}\] Note: This answer is typically written as one fraction (see the video), but that fraction is too long for some display sizes on our website. To solve this, we rewrote it as the difference of two fractions and used two lines. Both functions are equal.
2.
\(f\) is concave down on \((-\infty,0).\)
\(f\) is concave up on \((0,\infty).\)
Inflection Point: \((0,0)\)
\(f\) is concave up on \((0,\infty).\)
Inflection Point: \((0,0)\)
3.
Partition Numbers: \(x=2, \, 5, \, 7,\, 11\)
\(f\) is concave up on \((2,5)\) and \((11,\infty)\)
\(f\) is concave down on \((-\infty,2),\) \((5,7),\) and \((7,11)\)
Inflection points at \(x=2,\, 5,\, 11\)
\(f\) is concave up on \((2,5)\) and \((11,\infty)\)
\(f\) is concave down on \((-\infty,2),\) \((5,7),\) and \((7,11)\)
Inflection points at \(x=2,\, 5,\, 11\)
4.
A local maximum of \(-20\) at \(x=-10\)
A local minimum of 20 at \(x=10\)
A local minimum of 20 at \(x=10\)