Please help! the question is in the picture. how i do it?

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2 Answers
Apr 16, 2018

b. #f''# is never positive.

Explanation:

Notice that:

  • Near the left hand side of the curve, the tangent slopes up steeply,

  • As you traverse the curve from left to right, the tangent rotates until it slopes down steeply.

The slope is the derivative. Since it is decreasing from left to right, the second derivative is always negative.

Alternatively, note that the curve appears to match the definition:

#f(x) = 4-x^2#

Then:

#f'(x) = -2x#

and:

#f''(x) = -2#

Apr 16, 2018

#b# is correct.

Explanation:

.

This is a parabola that opens down. There are several ways to determine its equation. One of them is to look at the #x#-intercepts which are #x=-2 and 2#. This means:

#x-2=0# and #x+2=0#

Therefore, we can write the equation as:

#y=-(x-2)(x+2)#

#y=-(x^2-4)#

#y=-x^2+4#

The minus sign behind #x^2# is because the parabola opens down (in the negative direction).

#dy/dx=-2x#

#(d^2y)/(dx^2)=-2#

The second derivative is #=-2# and can not be positive for any values of #x#.