We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when $f'(x)>0$, $f(x)$ is increasing. The sign of the second derivative $f''(x)$ tells us whether $f'$ is increasing or decreasing; we have seen that if $f'$ is zero and increasing at a point then there is a local minimum at the point, and if $f'$ is zero and decreasing at a point then there is a local maximum at the point. Thus, we extracted information about $f$ from information about $f''$.

We can get information from the sign of $f''$ even when $f'$ is not zero. Suppose that $f''(a)>0$. This means that near $x=a$, $f'$ is increasing. If $f'(a)>0$, this means that $f$ slopes up and is getting steeper; if $f'(a)< 0$, this means that $f$ slopes down and is getting {\it less\/} steep. The two situations are shown in figure 5.4.1. A curve that is shaped like this is called concave up.

Figure 5.4.1. $f''(a)>0$: $f'(a)$ positive and increasing, $f'(a)$ negative and increasing.

Now suppose that $f''(a)< 0$. This means that near $x=a$, $f'$ is decreasing. If $f'(a)>0$, this means that $f$ slopes up and is getting less steep; if $f'(a)< 0$, this means that $f$ slopes down and is getting steeper. The two situations are shown in figure 5.4.2. A curve that is shaped like this is called concave down.

Figure 5.4.2. $f''(a)< 0$: $f'(a)$ positive and decreasing, $f'(a)$ negative and decreasing.

If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points. If the concavity changes from up to down at $x=a$, $f''$ changes from positive to the left of $a$ to negative to the right of $a$, and usually $f''(a)=0$. We can identify such points by first finding where $f''(x)$ is zero and then checking to see whether $f''(x)$ does in fact go from positive to negative or negative to positive at these points. Note that it is possible that $f''(a)=0$ but the concavity is the same on both sides; $\ds f(x)=x^4$ at $x=0$ is an example.

Example 5.4.1 Describe the concavity of $\ds f(x)=x^3-x$. $\ds f'(x)=3x^2-1$, $f''(x)=6x$. Since $f''(0)=0$, there is potentially an inflection point at zero. Since $f''(x)>0$ when $x>0$ and $f''(x)< 0$ when $x< 0$ the concavity does change from down to up at zero, and the curve is concave down for all $x< 0$ and concave up for all $x>0$.

Note that we need to compute and analyze the second derivative to understand concavity, so we may as well try to use the second derivative test for maxima and minima. If for some reason this fails we can then try one of the other tests.

Exercises 5.4

Describe the concavity of the functions in 1–18.

Ex 5.4.1 $\ds y=x^2-x$ (answer)

Ex 5.4.2 $\ds y=2+3x-x^3$ (answer)

Ex 5.4.3 $\ds y=x^3-9x^2+24x$ (answer)

Ex 5.4.4 $\ds y=x^4-2x^2+3$ (answer)

Ex 5.4.5 $\ds y=3x^4-4x^3$ (answer)

Ex 5.4.6 $\ds y=(x^2-1)/x$ (answer)

Ex 5.4.7 $\ds y=3x^2-(1/x^2)$ (answer)

Ex 5.4.8 $y=\sin x + \cos x$ (answer)

Ex 5.4.9 $\ds y = 4x+\sqrt{1-x}$ (answer)

Ex 5.4.10 $\ds y = (x+1)/\sqrt{5x^2 + 35}$ (answer)

Ex 5.4.11 $\ds y= x^5 - x$ (answer)

Ex 5.4.12 $\ds y = 6x + \sin 3x$ (answer)

Ex 5.4.13 $\ds y = x+ 1/x$ (answer)

Ex 5.4.14 $\ds y = x^2+ 1/x$ (answer)

Ex 5.4.15 $\ds y = (x+5)^{1/4}$ (answer)

Ex 5.4.16 $\ds y = \tan^2 x$ (answer)

Ex 5.4.17 $\ds y =\cos^2 x - \sin^2 x$ (answer)

Ex 5.4.18 $\ds y = \sin^3 x$ (answer)

Ex 5.4.19 Identify the intervals on which the graph of the function $\ds f(x) = x^4-4x^3 +10$ is of one of these four shapes: concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing. (answer)

Ex 5.4.20 Describe the concavity of $\ds y = x^3 + bx^2 + cx + d$. You will need to consider different cases, depending on the values of the coefficients.

Ex 5.4.21 Let $n$ be an integer greater than or equal to two, and suppose $f$ is a polynomial of degree $n$. How many inflection points can $f$ have? Hint: Use the second derivative test and the fundamental theorem of algebra.