Chapter 3, Problem 70RE

To determine

To justify: If two functions f and g have positive slopes at x =0, must the function f.g have a positive slope at x =0.

Answer to Problem 70RE

No, not necessarily. $F\text{'}\left(g\text{'}\left(x\right)\right)$ if $g\text{'}\left(x\right)=a$ positive number and $F\text{'}\left(x\right)$ where x is a positive number produces a negative number then $f\text{'}\left(g\text{'}\left(x\right)\right)$ would then be a negative number and therefore have a negative slope

Explanation of Solution

Given information: Two functions f and g have positive slopes at x =0.

Calculation:

  $\begin{array}{l}\text{Say}{F}^{\prime }(x)=-x^3+3,g^{\prime }(x)=3x+3.\\ x=0\text{then}{g}^{\prime }(0)=3\text{and}{F}^{\prime }(g^{\prime }(0))=-3^3+3=-24.\end{array}$

look at description and this is conclusive proof that $F\left(g\left(x\right)\right)$ does not mean positive slope (make sure that u understand that with positive slope we are talking about derivatives and not the actual f and g functions which is why we can use this justification).

No, not necessarily. $F\text{'}\left(g\text{'}\left(x\right)\right)$ if $g\text{'}\left(x\right)=a$ positive number and $F\text{'}\left(x\right)$ where x is a positive number produces a negative number then $f\text{'}\left(g\text{'}\left(x\right)\right)$ would then be a negative number and therefore have a negative slope