What x values is the function concave down if f(x)=15x23+5x?

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Janet Young · Accepted Answer

Step 1f(x)=15x23+5x is concave downward for all x&amp;lt;0As Kim suggested a graph should make this apparent (See bottom of this post).Alternately,Note that f(0)=0and checking for critical points by taking the derivative and setting to 0we getf′(x)=10x−13+5=0or10x13=−5which simplifies (if x&amp;lt;&amp;gt;0) tox13=−2→x=−8At x=−8f(−8)=15(−8)23+5(−8)=15(−2)2+(−40)Since (−8,20) is the only critical point (other than (0,0)) and f(x) decreases from x=−8 to x=0it follows that f(x) decreases on each side of (−8,20), sof(x) is concave downward when x&amp;lt;0When x&amp;gt;0 we simply note thata(x)=5x is a straight line andf(x)=15x23+5x remains a positive amount (namely 15x23 above that line therefore f(x) is not concave downward for x&amp;gt;0Graph {15x23+5x[−52,52,−26,26]}

soanooooo40 · Accepted Answer

Step 1Suppose that x=c is a critical point of f(x) then,If f′(x)&amp;gt;0 to the left of x=c and f′(x)&amp;lt;0 to the right of x=c then x=c is a local maximumIf f′(x)&amp;lt;0 to the left of x=c and f′(x)&amp;gt;0 to the right of x=c then x=c is a local minimum.If f′(x) is the same sign on both sides of x=c then x=c is neither a local maximum nor a local minimum.f′(x)=10x13+5Find intervals: Decreasing: −∞&amp;lt;x&amp;lt;−18, Increasing: −18&amp;lt;x&amp;lt;∞Plug x=−8 into 15x23+5x=20Minimum (−8,20)Plug x=0 into 15x23+5x=0Maximum (0,0)

What x values is the function concave down if f(x)=15x^{\frac{2}{3}}+5x?

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