How do you determine whether the function f(x)=5+12x−x3 is concave up or concave down and its intervals?

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Vegljamzt6 · Accepted Answer

Step 1Given:y=−x3+12x+5Find the first 2 derivativesdydx=−3x2+12d2ydx2=−6xSet the 1st derivative to zero to find for what value of &#039;x&#039; the curve turns.dydx⇒−3x2+12=0−3x2=−12x2=−12−3=4x=±4x=2x=−2At x=2 and x=−2 the curve turns. To determine whether it turns upwards or downwards, substitute the values in the 2nd derivative.At x=2; d2ydx2=−6x=−6×2=−12&amp;lt;0The curve has a maximum at x=2. In the immediate proximity the curve is concave downwards.At x=−2; d2ydx2=−6x=−6×−2=12&amp;gt;0The curve has a Minimum at x=−2. In the immediate proximity the curve is concave upwards.Point of inflection separates concavity from convexity. To the Point of inflection, set the 2nd derivative equal to zero.d2ydx2=0⇒−6x=0x=0At x=0, there is point of inflection.−∞&amp;lt;x&amp;lt;0; curve is concave upwards∞&amp;gt;x&amp;gt;0; curve is concave downwards.

How do you determine whether the function

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