Local max/min

[gymnast24]

Consider the function f(x), whose derivative f'=0 at x=2 and whose second derivative f''(x)= x^2+1. Choose one

a. f has a local maximum at x=2
b. f has a local minimum at x=2
c. f has a point of inflection at x=2
d. f has no critical numbers

So far I've set the second derivative to zero and got nothing so now I don't know what to do! I dont' get how the first derivative can be zero and the second derivative of the same function is x^2+1

 

[pka]

Check out this function: \(\displaystyle \L
\frac{{x^4 }}{{12}} + \frac{{x^2 }}{2} - \frac{{14x}}{3}\)

 

[soroban]

Hello, gymnast24!

Consider the function \(\displaystyle f(x)\), whose derivative \(\displaystyle f'\,=\,0\) at \(\displaystyle x\,=\,2\)
and whose second derivative is: \(\displaystyle f''(x)\:=\:x^2\,+\,1\)

a. \(\displaystyle f\) has a local maximum at \(\displaystyle x\,=\,2\,\)
b. \(\displaystyle f\) has a local minimum at \(\displaystyle x\,=\,2\)
c. \(\displaystyle f\) has a point of inflection at \(\displaystyle x\,=\,2\)
d. \(\displaystyle f\) has no critical numbers

So far I've set the second derivative to zero . . . why?
I dont' get how the first derivative can be zero . . .they plugged in x = 2
and the second derivative of the same function is x^2+1

They told us that the first derivative is zero when \(\displaystyle x\,=\,2\).
\(\displaystyle \;\;\)So there is a critical point there . . . a horizontal tangent.

It could be a maximum, a minimum, an inflection point, or none of these.
\(\displaystyle \;\;\)How do we find out? . . . the second derivative test.

Plug \(\displaystyle x\,=\,2\) into the second derivative: \(\displaystyle \,f''(2) \:=\:2^2\,+\,1\:=\:5\)
\(\displaystyle \;\;\)Since the second derivative is positive, the graph is concave up there: \(\displaystyle \cup\)

Therefore, there is a minimum at \(\displaystyle x\,=\,2\;\;\) . . . answer (b)

 

[pka]

Why not let the student discover the answer for her/himself??
There is no better way to kill learning than just to supply an answer.
The mathematics teacher that Keith Devlin has described as the greatest teacher is R.L.Moore. Prof. Moore forbid his student to read any outside text books. He required them use only his notes. He is arguably the most productive mathematics professor of the twentieth century. Three of his students have been president of the AMS and five of his students have been president of the MAA.
Discovery Learning just beats ‘Chalk an talk’ hands down?
It just as simple as the old tale “teach to fish or give a fish”.