# Answer to Question #17680 in Calculus for hsd

Question #17680

Let f(x)=(4x^(2))/(x^(2)+3)

Find the point(s) at which f achieves a local maximum.

Find the point(s) at which f achieves a local minimum.

Find the interval(s) on which f is concave up.

Find the interval(s) on which f is concave down.

Find all inflection points.

Find the point(s) at which f achieves a local maximum.

Find the point(s) at which f achieves a local minimum.

Find the interval(s) on which f is concave up.

Find the interval(s) on which f is concave down.

Find all inflection points.

Expert's answer

f'=(24 x)/(3+x^2)^2 f'=0 iff x=0

f''=-(72 (-1+x^2))/(3+x^2)^3 f"=0 iff x=1 or x=-1

x=0 we have local minimum because f'<0 if x<0 andf'>0 if x>0 there is no local maximumf">0 if x from(-1;1) so here f concave down

f"<0 at (-infinity;-1) and (1, infinity) so atthat interval it is concave up inflection points x=-1 x=1

f''=-(72 (-1+x^2))/(3+x^2)^3 f"=0 iff x=1 or x=-1

x=0 we have local minimum because f'<0 if x<0 andf'>0 if x>0 there is no local maximumf">0 if x from(-1;1) so here f concave down

f"<0 at (-infinity;-1) and (1, infinity) so atthat interval it is concave up inflection points x=-1 x=1

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