Calculate the speed of car for maximum economy

[chwala]

Homework Statement

see attached

Relevant Equations

differentiation

This is the problem,

This is the textbook solution ; i think the textbook solution is not correct!

My thought,
$S_{max}$ will be given by $$\frac {dy}{dS}=0$$
$$\frac {dy}{dS}=\frac {-1}{400}S+\frac {1}{5}=0$$
$$S=80$$
It is maximum economy because $y{''}<0$, i.e by use of the second derivative test.

 

[Mark44]

View attachment 294884

View attachment 294885
My thought,
$$S_{max}=\frac {dy}{dS}=0$$

This equation doesn't make sense, as it says that $S_{max} = \frac {dy}{dS}$ (not true) and that ##$S_{max} = 0$ (also not true).
I understand what you're trying to say, but this isn't the right way to say it.

$$\frac {dy}{dS}=\frac {-1}{400}S+\frac {1}{5}=0$$
$$S=80$$
It is maximum economy because $y{''}<0$, i.e by use of the second derivative test.

I get 80 (km/hr) as well.

 

[chwala]

This equation doesn't make sense, as it says that $S_{max} = \frac {dy}{dS}$ (not true) and that ##$S_{max} = 0$ (also not true).
I understand what you're trying to say, but this isn't the right way to say it.

I get 80 (km/hr) as well.

Let me re-phrase that...

 

[Mark44]

Let me re-phrase that...

Setting dy/dS to 0 is one way of finding extreme points, one of which may or may not be a maximum point.

 

[chwala]

Setting dy/dS to 0 is one way of finding extreme points, one of which may or may not be a maximum point.

That's why we use the second derivative test to ascertain that...Looks like textbook guys were after all wrong. Cheers Mark.

 

[Mark44]

That's why we use the second derivative test to ascertain that.

My point is that extreme points can be found at 1) points at which the derivative is 0; 2) endpoints of the domain, in case the domain is limited; 3) points at which the derivative doesn't exist, but the underlying function is defined.
Examples of cases 2 and 3:
$f(x) = \sqrt x$ (case 2) -- The derivative is never zero, but the function has a global minimum at x = 0.
$g(x) = |x|$ (case 3) -- The derivative is never zero, and fails to exist at x = 0, at which g has a global minimum.