Calculus Solutions EX #4.2

[embeddoc url=”” viewer=”google”] EXERCISE 4.2 3. (a) $$f(x)=3 x^{2}-6 x+1 \quad \quad (1)$$ First derivative test Differentiating (1) w.r.t $x$. $$ f^{\prime}(x)=6 x-6 \quad \quad (2) $$ Given that $x_{0}=1.$ Sign analysis of $f^{\prime}(x)$ $$ \begin{center} \begin{tabular}{|c|c|c|c|}\hline\text {Interval} & $\begin{array}{c}\text { test point } \\ c\end{array}$ & $\begin{array}{l}f^{\prime }(c) \end{array}$ & Sign of $f^{\prime}(x)$ \\\hline$x<1$ … Read more

Calculus Solution Ex # 4.5

  Exercise 4.5 Question 1 Let $x$ be the required number, then the reciprocal is $\frac{1}{x}$. $$f(x)=x+\frac{1}{x};\quad\left[\frac{1}{2}, 3 / 2\right]——–(1)$$ differentiating (1) w.r.t $x$ $$ f^{\prime}(x)=1-\frac{1}{x^2} $$ Set $$f^{\prime}(x)=0$$ $$ \Rightarrow 1-\frac{1}{x^2}=0 \Rightarrow 1=\frac{1}{x^2}$$ $$ \Rightarrow x^2=1, \quad \Rightarrow x= \pm 1 $$ But only $x=1$ lie in the interval $\left(\frac{1}{2}, \frac{3}{2}\right)$, therefore neglecting $x=-1$. … Read more

Calculus solution Ex#4.4

  Exercise 4.4 Question 7 Find the absolute maximum and minimum values $f(x)=4 x^2-12 x+10 ;[1,2]$ Solution Since it is a polynomial, therefore its continuous and differentiable everywhere. To find critical points: $$f(x)=4 x^2-12 x+10——-(1)$$ differentiating w.r.t $x$ $$ f^{\prime}(x)=8 x-12 $$ Set $f^{\prime}(x)=0 $ $$ \Rightarrow 8 x-12=0 \Rightarrow 8 x=12$$ $$ \Rightarrow x=\frac{12}{8}=\frac{3}{2} … Read more

Calculus Solutions EX#4.1

EXERCISE 4.1 10.  (a)  $f$ is increasing on $(-\infty,+\infty)$ (b)  $f$ is not decreasing anywhere. (C) $f$ is concave up on $(-\infty, 1)$ $\quad$ and $(3,+\infty)$ (d)  $f$ is concave down $(1,3)$ (e) the $x$-coordinates of all inflection points $x=1$  and $x=3 $ 15-32. Find (a) the intervals on which $f$ is increasing, (b) open … Read more