#Please Help With Calculus Proof.
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gentle eagle
You know that f'', g'' < 0, right?
pseudo veldt
You know that `f'', g'' < 0`, right?
Yes because questions tells us that f and g are concave down.
gentle eagle
That's what it means for a function to be concave down. You also know f is increasing, which means what?
pseudo veldt
Not too sure.
gentle eagle
If f is increasing, f'...?
pseudo veldt
f'(x1) < f'(x2) ?
pseudo veldt
Positive
gentle eagle
...well, nonnegative, technically.
pseudo veldt
Ok so f'(x) is >= 0
gentle eagle
Right. If f were strictly increasing, which I think might be what the question is actually asking, then f'(x) > 0.
gentle eagle
...you absolutely sure, or are you just assuming?
pseudo veldt
I'm sure because otherwise it would have said non-decreasing
gentle eagle
Okay, then.
pseudo veldt
Since f is concave down which means f''(x) < 0, does that mean f''(g(x)) is also < 0?
gentle eagle
Is that relevant to this question?
It could be. An instance where the domain of f'' didn't contain the range of g (but the domain of f did) would qualify as a counterexample. It would mean h'' was undefined, not negative.
Though I'm not entirely sure such an instance exists or how to find it.
gentle eagle
I'm sure because otherwise it would have said non-decreasing
The question actually just says no derivative is equal to zero.
pseudo veldt
The question actually just says no derivative is equal to zero.
So that means f'(x) > 0
gentle eagle
So that means f'(x) > 0
Or undefined.
pseudo veldt
Since f''(g(x)) < 0 and is being multiplied by a positive which is added to product of a positive and negative shows that h''(x) < 0 as required.
gentle eagle
Assuming all the functions involved are defined, yes.
pseudo veldt
Ok cool
I can't think of any case where they wouldn't be defined.
Let me know if you find any.
gentle eagle
Let me know if you find any.
Something with f''(x) = ln(x) and g(x) <= 0?
Wait, duh. h can't be concave down if it's undefined. f(x) = ln(x) g(x) = -x^2
gentle eagle
Wait, duh. `h` can't be concave down if *it's* undefined.```
f(x) = ln(x)
g(x) =...
@pseudo veldt
pseudo veldt
,w y intercept x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w asymptotes x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w crtical points x\left(\sqrt[3]{\left(\ln\left(x^{2}\right)^{2}\right)}\right)
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pseudo veldt
,w intervals concave up f\left(x\right)\ =\ x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
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Requested by ASRK1#5890
pseudo veldt
,w intervals concave up \ x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
Requested by ASRK1#5890
pseudo veldt
,w concave up x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w points of inflection x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w increasing x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w concave down x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
,w decreasing x\left(\ln\left(x^{2}\right)\right)^{\frac{2}{3}}
candid slateBOT
pseudo veldt
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