# 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.

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}