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Question

# (a) Explain why the Intermediate Value Theorem gives no information about the zeros of the function $f(x)=\ln \left(x^{2}+2\right)$ on the interval [-2, 2]. (b) Use technology to determine whether f has a zero on the interval [-2,2].

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We would like to answer of the following question for the following expression.

$\color{Brown}\ f(x)=\ln\ (x^2+2)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{At interval},\ [-2,\ 2]$

$\lozenge$\ \ $\text{\underline{\bf{Solution:}}}$

$\text{\color{#4257b2}(a)\ Explain why the intermediate value theorem gives no information about zeros of the function at the indicated interval.}$

First thing we will do, get the domain of the given function. this function is logarithmic function, so the domain equal to all real number.

Now we determine the value of function at the indicated interval as follows:

$f(-2)=\ln\ ((-2)^2+2)=\ln\ (4+2)=\ln\ (6)=1.79\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boxed{\ f(-2)=1.79\ }$

$f(2)=\ln\ ((2)^2+2)=\ln\ (4+2)=\ln\ (6)=1.79\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boxed{\ f(2)=1.79\ }$

From the above calculation, we have a positive value at \ $(x=-2,\ x=2)$,\ so the intermediate value theorem cant gives any information about zeros.

$\text{\color{#4257b2}(b)\ Use technology to determine whether the function (f) has or not zeros at the indicated interval.}$

According the result from part $(a)$\ we can say that the function has no zero at the indicated interval

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