Question

need hw help

For the function f(x) = ln(x2+25). Find all interval(s) where f(x) is increasing. Select all that apply. (O, 5) (-0, 0) (-5, 0) ol-00,-5) None (0, 2) (5, 00)

Answer 1

Solution:

Given function is

f(1) = ln(22 +25)

Differentiate with respect to
T
, we get

f'(2) = 2.c x2 + 25

Again Differentiate with respect to
T
, we get

$f''(x)=\frac{2(x^2+25)-2x(2x)}{(x^2+25)^2}=\frac{2x^2+50-4x^2}{(x^2+25)^2}$

$f''(x)=\frac{50-2x^2}{(x^2+25)^2}$

a). is increasing if

Thus,   is increasing if
T
.

Hence,

is increasing on $(0,5),(0,\infty),(5,\infty)$ .

Which are the required intervals.

b).   is decreasing if

Thus,   is decreasing if .

Hence,

is decreasing on $(-\infty,0),(-\infty,-5)$ .

Which are the required intervals.

c). Now critical points is given by

f'(c) = 0

$\Rightarrow \frac{2x}{x^2+25}=0$

$\Rightarrow x=0$

Now,

$f''(0)=\frac{50}{(25)^2}>0$

Thus, $x=0$ is a point of minima.

And the minimum value is given by

$f(0)=ln(25)=ln(5^2)=2ln(5)$

Hence, is a relative minimum point.

d). Now, cut points of are given by

f"(C) = 0

$\Rightarrow \frac{50-2x^2}{(x^2+25)^2}=0$

$\Rightarrow 50-2x^2=0$

$\Rightarrow x^2=25$

$\Rightarrow x^2=5,-5$

Which are the required cut points.

This complete the solution.