# Let F\left ( x \right ) = \int_{9}^{x}\frac{1}{ln\left ( 4t \right )}dt, \ for \ x\geq 9 . a....

## Question:

Let {eq}F\left ( x \right ) = \int_{9}^{x}\frac{1}{ln\left ( 4t \right )}dt, \ for \ x\geq 9 {/eq}.

a. F?(x) = _____

b. On what interval or intervals is increasing?

c. On what interval or intervals is the graph of concave up?

## First and Second Derivatives:

With the first derivative we identify critical points, also intervals of increase and decrease of the function, in addition, with the second derivative we identify points of inflection and concavity.

## Answer and Explanation:

We have,

{eq}F\left ( x \right ) = \int_{9}^{x}\frac{1}{ln\left ( 4t \right )}dt \\ {/eq}

Leibniz formula for integrals:

{eq}f'(x)= \displaystyle \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x))* \frac{dh}{dx} - f(g(x))* \frac{dg}{dx} \\ {/eq}

a.

Applying the Leibniz formula:

{eq}f'(x)= \displaystyle \int_{9}^{x}\frac{1}{ln\left ( 4t \right )}dt = \frac{1}{ \ln(4x) }* 1 - 0 \\ f'(x)= \frac{1}{ \ln(4x) } \\ {/eq}

b.

{eq}f'(x)=DNE {/eq} when {eq}x=0 {/eq}

b.

{eq}\begin{array}{r|D{.}{,}{5}} Interval & {0<x<\infty} \\ \hline Test \space{} value & \ x=5 \\ Sign \space{} of \ f'(x) & \ f'(5)>0 \\ Conclusion & increasing \\ \end{array} \\ {/eq}

c.

Differentiating the function

{eq}f''(x)=-\frac{ 1}{ x( \ln(x)+2 \ln(2) )^{2} } \\ {/eq}

{eq}f''(x)=0 {/eq} The function has no real solution, therefore, the function has no inflection point.

{eq}\begin{array}{r|D{.}{,}{5}} Interval & {0<x<\infty } \\ \hline Test \space{} value & \ x=5 \\ Sign \space{} of \ f'' (x) & \ f'' (5)<0 \\ Conclusion & concave \space down \\ \end{array} \\ {/eq}

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from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9