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The second derivative of a function
f is given by

f^('')(x)=sin(x^(2))+cos(2x)-(1)/(2)". "
How many points of inflection does the graph of
f have on the interval
1 < x < 4 ?
Use a graphing calculator.
Choose 1 answer:
(A) Two
(B) Three
(C) Four
(D) Five

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

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Find higher derivatives of rational and radical functions

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

(C) Four

\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.

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