The second derivative of a function $f$ is given by $\newline$ $f^{\prime \prime}(x)=\sin \left(x^{2}\right)+\cos (2 x)-\frac{1}{2} \text {. }$ $\newline$ How many points of inflection does the graph of $f$ have on the interval $1<x<4$ ? $\newline$ Use a graphing calculator. $\newline$ Choose $1$ answer: $\newline$ (A) Two $\newline$ (B) Three $\newline$ (C) Four $\newline$ (D) Five

Full solution

Q. The second derivative of a function

f

is given by

\newline

f^{\prime \prime}(x)=\sin \left(x^{2}\right)+\cos (2 x)-\frac{1}{2} \text {. }

\newline

How many points of inflection does the graph of

f

have on the interval

1<x<4

\newline

Use a graphing calculator.

\newline

Choose

1

answer:

\newline

(A) Two

\newline

(B) Three

\newline

\newline

(D) Five

Find Points of Inflection: To find points of inflection, we need to look at where the second derivative changes sign.
Plot Second Derivative: Use a graphing calculator to plot $f''(x) = \sin(x^2) + \cos(2x) - \frac{1}{2}$ on the interval $1 < x < 4$ .
Identify Potential Points: Identify the $x$ -values where $f''(x)$ crosses the $x$ -axis; these are potential points of inflection.
Count Sign Changes: Count the number of times $f''(x)$ changes sign around these $x$ -values.
Determine Points of Inflection: If $f''(x)$ changes sign from positive to negative or negative to positive, then that $x$ -value is a point of inflection.
Analyze Sign Changes: After using the graphing calculator, it looks like $f''(x)$ changes sign three times within the interval.