For what values of x is #f(x)= 7x^3 + 2 x^2 + 7x -2 # concave or convex?

Answer 1

Concave down for #x < -2/21#, concave up for #x > -2/21#

First, we can try to find inflection points for this function. An inflection point is a point where the concavity changes, so finding this point is often helpful when analyzing concavity.

The process is straightforward; we will find #x# such that #d^2/dx^2 f(x) = 0#, that is, when the second derivative of #f# is zero.

Start by finding the 1st derivative, by simply applying the power rule to each term:

#d/dx f(x) = 21x^2 + 4x + 7#

Then, differentiate again to find the 2nd derivative:

#d^2/dx^2 f(x) = 42x + 4#

So, now we set the thing equal to zero:

#0 = 42x + 4#
And solve for #x#:
#x = -2/21#
So now we know that the function has one inflection point. What about the rest of possible values for #x#?

Well, if a segment of a graph is concave up (its slope is increasing) then the 2nd derivative will be positive. And if a segment is concave down, with a decreasing slope, the 2nd derivative will be negative.

Since we know that the 2nd derivative switches from negative to positive or vice versa at #x = -2/21#, let's see whether it's positive or negative, at, for instance, #x = 0#:
#d^2/dx^2 f(0) = 42*0 + 4 = 4#
Interesting. So the 2nd derivative is positive at #x=0#. This tells us that, since the only switch occurs at #x = -2/21#, all #x > -2/21# have a positive 2nd derivative as well, and are therefore concave up.
On the other hand, all #x < -2/21# are concave down, since the switch must occur at #x=-2/21#.

Hopefully this makes sense.

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Answer 2
To determine the concavity of the function \(f(x) = 7x^3 + 2x^2 + 7x - 2\), we need to find its second derivative and then analyze its sign. The first derivative of \(f(x)\) is \(f'(x) = 21x^2 + 4x + 7\). The second derivative of \(f(x)\) is \(f''(x) = 42x + 4\). To find the values of \(x\) for which the function is concave up (convex) or concave down (concave), we need to find the roots of \(f''(x)\) and then analyze the sign of \(f''(x)\) in the intervals determined by these roots. Setting \(f''(x) = 0\) gives us \(42x + 4 = 0\), which simplifies to \(42x = -4\) and then \(x = -\frac{2}{21}\). Since the coefficient of \(x\) in \(f''(x)\) is positive (42), the function is concave up (convex) for \(x < -\frac{2}{21}\) and concave down (concave) for \(x > -\frac{2}{21}\).
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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