1.60 iano \( f(x), g(x) \) due funzioni derivabili per \( x>a \). Dimostrare che \( f(x) \leq g(x) \), per ogni \( x>a \), se valgono le due condizioni: \[ f^{\prime}(x) \geq g^{\prime}(x), \quad \forall x>a ; \quad \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 \text {. } \] [Applichiamo il teorema di Lagrange alla differenza \( f-g \) nell' intervallo \( [x, b] \), con \( a<x<b \). Esiste \( \xi \in(x, b) \) per cui \( [f(b)-g(b)]-[f(x)-g(x)]=\left[f^{\prime}(\xi)-g^{\prime}(\xi)\right](b-x) \geq 0 \) Al limite, per \( b \rightarrow+\infty \), otteniamo la tesi \( ] \)

Math 60 Exam \#4 1) Use Right-endpoint Rule with \( n=5 \) to approximate \( \int_{2}^{3} x \sqrt{x+1} d x \) to 4 decimal places.

II. Dadas las funcione \( \mathrm{f}(\mathrm{x})=5 \mathrm{x}^{2}+4 \mathrm{x}-1 \) \( \mathrm{~g}(\mathrm{x})=\mathrm{x}^{2}-6 \) Encontra a) \( \operatorname{Lim}_{x \rightarrow 1}\left[\frac{f(x)}{g(x)}\right] \) b) \( \operatorname{Lim}_{x \rightarrow 0}[f(x) \cdot g(x)] \)

Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) At which zeros does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=\square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around.

Find the average rate of change of the function \( f(x)=x^{2}+9 x \) from \( x_{1}=1 \) to \( x_{2}=2 \). The average rate of change is \( \square \). (Simplify your answer.)

Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) A. \( x=\square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c) Find the \( y \)-intercept by computing \( f(0) \). (0) \( =\square \) d) Determine the symmetry of the graph. Odd; origin symmetry