4- Dada a equação \( f(x)=x^{3}+x-1 \), tal que \( E_{A}<0.001 \) e \( V F(10,4,-6,6) \) a) Avalie o intervalo onde se encontra a raiz da equação graficamente e pá método de tabelamento e comente o resultado ( 3 val)

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal mintmum value at \( x=1 \). what is the value of \( a \) and \( c \) ?

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal manamum value at \( x=1 \), + What is the value of a \( f c \) ?

Find the volume of the solid formed by rotating the area under the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 3 \) about the y-axis.

Evaluate the triple integral. \[ \iiint_{G} x y \sin (y z) d V \text {, } \] where \( G \) is the rectangular box defined by the inequalities \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)

Evaluate the following triple integral in spherical coordinates. \[ f(\rho, \theta, \varphi)=\sin (\varphi) \] over the region \( 0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \frac{\pi}{4}, \quad 7 \leq \rho \leq 8 \)