For what values of x is $f(x)=(5x-1)(x-5) (2x+3)$ concave or convex?

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Answer 1 · 2018-01-13T12:26:37

Convex $color(white)(888)x in(37/30 , oo)$

Concave $color(white)(888)x in (-oo , 37/30)$

We can test for concavity using the second derivative. If:

$(d^2y)/(dx^2)>0$ convex ( concave up )

$(d^2y)/(dx^2)<0$ concave ( concave down )

$(d^2y)/(dx^2)=0$ concave/convex or point of inflection. This would have to be tested.

$f(x)=(5x-1)(x-5)(2x+3)$

It will make the differentiation easier if we expand this:

$(5x-1)(x-5)(2x+3)=10x^3-37x^2-68x+15$

The second derivative is the derivative of the first derivative, so:

$dy/dx(10x^3-37x^2-68x+15)=30x^2-74x-68$

$(d^2y)/(dx^2)=dy/dx(30x^2-74x-68)=60x-74$

$:.$

$60x-74>0$ , $x>37/30$

Convex $color(white)(888)x in(37/30 , oo)$

$60x-74<0$ , $x<37/30$

Concave $color(white)(888)x in (-oo , 37/30)$

GRAPH:

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For what values of x is f(x)=(5x-1)(x-5) (2x+3)f(x)=(5x-1)(x-5) (2x+3) concave or convex?