Question #1429

For y= x^(4)-72x^(2)-17 use analytic methods to find the exact intervals on which the function is..<br>
A)increasing<br>
B)decreasing<br>
C)concave up<br>
D)concave down<br>
E)find the local extreme values<br>
F)find the inflection points

Expert's answer

Let's find the first and the second derivatives of the function:

y' = 4x3 - 144x = 0; y' = 0: x1 = 0; x2 = -12; x3 = 12

y'' = 12x2 - 144 = 0; y'' = 0: x1 = -2√(3); x2 = 2√(3)

So the local extreme points are {-12,0,12}

Inflection points {-2√(3), 2√(3)}

The function increases when y' > 0 and decreases when y' < 0.

Thus on the intervals (-inf,-12) and (0,12) the function decreases, on (-12,0) and (12, inf) it increases.

On the intervals (-inf, - 2√(3)) and (2√(3), inf) it concaves down, on (-2√(3), 2√(3)) - up.

y' = 4x3 - 144x = 0; y' = 0: x1 = 0; x2 = -12; x3 = 12

y'' = 12x2 - 144 = 0; y'' = 0: x1 = -2√(3); x2 = 2√(3)

So the local extreme points are {-12,0,12}

Inflection points {-2√(3), 2√(3)}

The function increases when y' > 0 and decreases when y' < 0.

Thus on the intervals (-inf,-12) and (0,12) the function decreases, on (-12,0) and (12, inf) it increases.

On the intervals (-inf, - 2√(3)) and (2√(3), inf) it concaves down, on (-2√(3), 2√(3)) - up.

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