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Answer 1

Răspuns:

Explicație pas cu pas:

1) f'(x) = (2x(x+1) -x^2*1)/(x+1^2 = (x^2 +2x)/(x+1)^2

f'(1) = (1+2)/(1+1)^2 = 3/4

f''(x) =[(2x+2)(x+1)^2 -(x^2+2x)*2(x+1)*1)]/(x+1)^4

Calculezi f''(2)...

b) f(x) nu e definita in x = -1, aici avem, asimpt. verticala

lim x->-inf (f(x) = -inf

(grad numarator > grad numitor, si numitorul < 0)

lim x->+inf (f(x) = +inf

f'(x) =0,  x^2+2x = x(x+2) = 0

x1 = -2,  x2 = 0

lim st x->-1 = -inf,  lim dr x->-1 = +inf

f(-2) = 4/-1 = -4,  f(0) = 0

(-2; -4) punct max.,   (0;0) punct de minim

x in (-inf, -2) crescatoare

x in (-2, -1) descrescatoare

x in (-1, 0)   descrescatoare

x in (0, +inf)  crescatoare

2) f(x) = x +1/e^x

a) f'(x) = 1 -e^x/(e^x)^2 = 1 -1/e^x = 1 - e^(-x)

b) f'(x0 = 0, (e^x -1)/e^x = 0, e^x = 1 =e^0,  x = 0

lim x-> -inf (f(x)) = +inf,  daca x < 0,  -x >0,

iar e^x creste mult mai repede decat x

lim x-> +inf (f(x)) = +inf +0 = +inf

f(0) = 0 +1/e^0 = 0+1 = 1

(0; 1) punct de minim

x in (-inf, 0) descrescatoare

x in (0, +inf)  crescatoare