Answer to Question #275079 in Calculus for Leonardo

Question

The derivative of a differentiable function ff(xx) is given as

ff′

(xx) = xx + 3

(xx − 2)2 .

a. Find intervals of increase and decrease for ff(xx).

b. Determine values of xx for which relative maxima and minima occurs on the graph of

ff(xx).

c. Find ff′′(xx) and determine intervals of concavity for the graph of ff(xx).

d. For what values of xx do inflection points occur on the graph of ff(xx).

Answer 1

$$f'(x)=x+3(x-2)^2$$

a.

Let domain of $f(x)$ is $(-\infin, \infin)$

Find the critical number(s)

$$f'(x)=0=>x+3(x-2)^2=0$$

$$3x^2-11x+12=0$$

$$D=(-11)^2-4(3)(12)=-23<0$$

There is no any critical number.

$f'(x)>0$ for $x\in (-\infin, \infin).$

The function $f(x)$ increases on $(-\infin, \infin).$

The function $f(x)$ is never decreases.

b. There are neither relative maxima nor relative minima.

c.

$$f''(x)=(x+3(x-2)^2)'=1+6(x-2)$$

$$=6x-11$$ $$f''(x)=0=>6x-11=0=>x=\dfrac{11}{6}$$

If $x<\dfrac{11}{6}, f''(x)<0,f(x)$ is concave down.

If $x>\dfrac{11}{6}, f''(x)>0,f(x)$ is concave up.

The graph of $f(x)$ is concave up on $(\dfrac{11}{6}, \infin).$

The graph of $f(x)$ is concave down on $(-\infin,\dfrac{11}{6}).$

d. The inflection point occurs at $x=\dfrac{11}{6}.$