I have given the function and it asks me to indicate the points of inflection and where the function is concave up(CU) or concave down(CD);
I proceded with the first derivative f'(x)=e^(13x)*x(13x+2),
- the second f"(x)=e^(13x)*[169x^(2)+52x+2],
since e^x is never 0, this one gives me[169x^(2)+52x+2]=0 for x=[-(52)+(sqrt(1352))]/(2*169) and x=[-(52)-(sqrt(1352))]/(2*169);
it bugs that those points are quite bizarre and on all the intervals the function gives me CU, does that mean that I don't have inflection points here, I might have mistaken something...
Just one question -
do we have to evaluate the second derivative on the first derivative's critical points; I have seen the examples where we find the critical points for the second derivative and evaluate it on them.... I'm confused...
The steps are as under:
Step 1. find the derivative f'(x)
Step-2. Equate f'(x) = 0 and solve. These are the values of x where the function may have a maxima ( CD) or Minima ( CU)
Step 3. Find f"(x)
Step 4. Find the value of f"(x) for the values of x found in step 2.
step 5. The values of x for which f"(x) is < 0 the function will have a maxima ( CD) and the values of x for which f"(x) > 0 the function will have a minima ( CU).
Step 6. If one is to find the maximum or minimum value then we have to evaluate f(x) at those values of x [ Step - 2 ]
but well, I have to find the 2 points(A and B) of inflection and whether the function is CU or CD on the intervals (-Inf, A), (A, B), (B, Inf) ,
and I found the critical points for the second derivative: x=-0.045060495(B) and x=-0.262631812 (A),
finally, the function would be CU -> CD -> CU
The answers are correct as I found them, thanks for help... though the previous advises weren't quite precise
hallsofivy, I thought I was pretty clear in my first post, but I'm especially thankful for your critical posts, I like them!