Section 4.4 Concavity and Curve Sketching
Consider the graph of \(y=x^3\text{.}\) Concavity is how we describe the difference in how the graph bends differently to the left and right of the origin.
Definition 4.4.1. Concavity.
The graph of a differentiable function \(y=f(x)\) is
concave up on an open interval \(I\) if \(f'\) is increasing on \(I\text{.}\)
concave down on an open interval \(I\) if \(f'\) is decreasing on \(I\text{.}\)
Fact 4.4.2. Second derivative test for concavity.
Let \(y=f(x)\) be a twice differentiable function on an interval.
If \(f''\gt 0\) on the interval, then \(f\) is concave up.
If \(f''\lt 0\) on the interval, then \(f\) is concave down.
Example 4.4.3.
\(\displaystyle\frac{d^2}{dx^2} x^3 = 6x\text{;}\) so \(y=x^3\) is concave down on \((-\infty,0)\) and concave up on \((0,\infty)\text{.}\)
\(\displaystyle\frac{d^2}{dx^2}x^2=2\text{,}\) so \(y=x^2\) is concave up on \((-\infty,\infty)\text{.}\)
Definition 4.4.4. Point of inflection.
A point at which the graph of a function has a tangent line (possibly vertical) and where the concavity changes is called a point of inflection.
Fact 4.4.5.
If a function has a point of inflection at \(x=c\text{,}\) then either
\(f''(c)=0\text{,}\) or
\(f''(c)\) does not exist.
Example 4.4.6.
\(f(x)=x^3\text{:}\) \(f''(0)=0\text{;}\) has point of inflection.
\(g(x)=x^4\text{:}\) \(g''(0)=0\text{;}\) has no point of inflection.
\(h(x)=x^{5/3}\text{:}\) \(h''(0)\) does not exist; has point of inflection.
\(k(x)=x^{4/3}\text{:}\) \(k''(0)\) does not exist; has no point of inflection.
Fact 4.4.7. Second derivative test for local extrema.
Let \(f\) be a function such that \(f''\) is continuous on an open interval around \(x=c\text{.}\)
If \(f'(c)=0\) and \(f''(c)\lt 0\text{,}\) then \(f\) has a local minimum at \(x=c\text{.}\)
If \(f'(c)=0\) and \(f''(c)\gt 0\text{,}\) then \(f\) has a local maximum at \(x=c\text{.}\)
If \(f'(c)=0\) and \(f''(c)= 0\text{,}\) then the test gives no information; there may or may not be a local extremum.
Fact 4.4.8. Curve Sketching.
To sketch \(y=f(x)\text{:}\)
Identify the domain
Identify any obvious symmetry (e.g., even/odd functions)
Find \(y'\) and \(y''\)
Find the critical points
Identify local extrema and intervals on which the function is increasing and decreasing
Find potential points of inflection
Find the intervals on which the function is concave up and down and the locations of points of inflection
Identify all vertical and horizontal or oblique asymptotes
Plot asymptotes and key points---\(y\)-intercept; \(x\)-intercepts when possible and convenient; local extrema; points of inflection; end points of the domain---and skecth
Example 4.4.9.
Sketch the graph of:
\(\displaystyle y=x^3(x+2)\)
domain is \((-\infty,\infty)\)
no obvious symmetry
-
\(y'=x^2(4x+6)\)
critical points at \(x=0\text{,}\) \(x=-\frac32\)
decreasing on \((-\infty,-\frac32)\text{,}\) increasing on \((-\frac32, \infty)\)
global minimum at \(\left(-\frac32,-\frac{27}{16}\right)\text{;}\) no other extrema
-
\(y'' = 12x(x+1)\)
potential points of inflection at \(x=-1\) and \(x=0\)
concave up on \((-\infty,-1)\) and \((0,\infty)\text{;}\) concave down on \((-1,0)\)
points of inflection at \((-1,-1)\) and \((0,0)\)
No asymptotes
Example 4.4.10.
Sketch the graph of:
\(\displaystyle y=x^2+\frac{2}{x}\)
\(\displaystyle y=5x^{2/5}-2x\)
\(y = \sqrt{3}x-2\cos x\text{,}\) \(-\pi\leq x\leq \pi\)