Points of inflection Find the x-coordinate of the point(s) of ...

Question

Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says determine the X coordinates of the points of inflection of F of X, which is equal to 2, multiplied by the hyperbolic tangent square rod of X plus 1. So we need to determine the X coordinates for our inflection points of our function F of X. So the first step is to find the second derivative of our function. So starting off by finding the first derivative, which is going to be F of X, this is going to be equal to the derivative with respect to X. Of the quantity of 2 multiplied by the hyperbolic tangent squared of X plus 1 in quantity. So taking this derivative, this is going to be equal to. The 2 is a constant, so you can move through the derivatives we'll have 2 multiplied by the derivative with respect to X of the hyperbolic tangent squared of X. So we'll have to first apply our power rule. So we'll have 2 multiplied by the hyperbolic tangent of X. But we'll need to apply our chain rule and multiply this by the derivative with respect to x of the hyperbolic tangent of X, which, if you recall, is going to be equal to the hyperbolic sequent squared of X. And then we'll have plus the derivative of 1 with respect to X, which is 0. So simplifying this expression, this is going to be equal to 4 multiplied by the hyperbolic tangent of X, multiplied by the hyperbolic sequent squared of X. So next we need to find the second derivative, which is going to be F of X, and this is going to be equal to the derivative of our first derivative. So we'll have the derivative with respect to X. Of 4 multiplied by the hyperbolic tangent of X, multiplied by the hyperbolic sequent squared of X. So taking this derivative, this is going to be equal to. Again, our constant of 4 can move through the derivative, so we'll have 4 multiplied by the quantity. And here we have a product, so we're going to apply a product rule. So we call for the product rule that we have the derivative of the first term. So the derivative with respect to X of the hyperbolic t of X, which we saw earlier was the hyperbolic sequent squared of X, and then this this gets multiplied by the second term in our product, which is also the hyperbolic. Sequence squad of x. Then we'll have plus the first terminal product, which is the hyperbolic tangent of X. Then this gets multiplied by the derivative of the second terminal product, so we'll have the derivative with respect to X of the hyperbolic sequence square of X. So applying our power rule first, we'll have 2 multiplied by the hyperbolic sequent of X. And this needs to be multiplied by the derivative with respect to X of the hyperbolic. of X, since we need to apply our chain rule and recall that the derivative with respect to X of the hyperbolic sequent of X is minus the hyperbolic tangent of X multiplied by the hyperbolic sequant of X in quantity. So now we just need to simplify this expression. So this is going to be equal to, and we noticed that we can factor out a hyperbolic sequent squared of X out of both terms. So we'll have 4 multiplied by the hyperbolic sequent squared of X, multiplied by the quantity of the hyperbolic sequent squared of X minus 2 multiplied by the hyperbolic tangent squared of X in quantity. Now recall that we can rewrite the hyperbolic sequence square of X as being equal to 1 minus the hyperbolic tangent square rod of X. So using that identity, this is going to be equal to 4 multiplied by the hyperbolic sequence squared of X, multiplied by the quantity of 1 minus 3 multiplied by the hyperbolic tangent squared of X in quantity. So now we need to find our inflection points. And recall that possible inflection points occur whenever our second derivative here is equal to zero or undefined. Now, the hyperbolic sequin of X and the hyperbolic tangent of X is defined for all values of X. So therefore, FX is defined for all values of X. So the only time we're going to get possible inflection points is when our second derivative here is equal to 0. And recall that the hyperbolic sequent of X can never be equal to 0, so the only time that our second derivative here can be equal to 0. is when the quantity of 1 minus 3 multiplied by the hyperbolic tangent square of X in quantity is equal to 0. So we're going to set 0 equal to 1 minus 3 multiplied by the hyperbolic tangent squared of X and solved for X. So the first step is to add 3 multiplied by the hyperbolic tangent squared of Xx to both sides. So that will give us 3 multiplied by the hyperbolic tangent squared of X is equal to 1. Dividing both sides of the equation by 3, we'll get that the hyperbolic tangent squared of X is equal to 1 divided by 3. Taking the square root of both sides, we'll get that the hyperbolic tangent of X is equal to plus or minus the square root of 3 divided by 3. And then taking the inverse hyperbolic tangent of both sides of the equation, we'll get that X is equal to plus or minus the inverse hyperbolic tangent. Of the square root of 3 divided by 3. So, here we have used the identity that the inverse hyperbolic tangent of minus X is equal to minus the inverse hyperbolic tangent of X. So we have 2 possible X values to check to see if we actually have inflection points, and so we need to determine if The second derivative actually changes sine at these points. So, we're gonna do this by evaluating this expression or 3 values of Xs. So, the first value that we're going to look at. is when X is equal to -1. Since the inverse. Hyperbolic tangent of the square root of 3 divided by 3 is approximately equal to 0.66. So X is going to be plus or minus 0.66. So choosing X equal to -1 means that we've chosen an x value that is less than X equal to -0.66. So, in this case, we'll need to evaluate F of -1. And this is going to be equal to. 4 multiplied by the hyperbolic sequence squared of minus 1 multiplied by the quantity of 1 minus 3 multiplied by the hyperbolic tangent squared of minus 1 in quantity. So, evaluating this expression using our calculators, we see that this is going to be approximately equal to -1.24. So the next value that we want to look at for X is when X is going to be equal to 0. And so this is going to be a value that is greater than x equal to minus 0.66, but less than X equal to 0.66. So here we're going to have F of 0 that we need to calculate, and so this is going to be equal to 4 multiplied by the hyperbolic sequence squared of 0, multiplied by the quantity of 1 minus 3 multiplied by the hyperbolic tangent squared. Of 0 in quantity. And evaluating this expression, we see that this is going to be equal to 4. And for the final value of x that we're going to look at, that's going to be X equal to 1, since this is going to be greater than x equal to 0.66. And so calculating F of 1, we see that this is going to be equal to 4 multiplied by the hyperbolic degan squared of 1, multiplied by the quantity of 1 minus 3, multiplied by the hyperbolic tangent squared. Of 1 in quantity, and when we evaluate this expression, this is going to be approximately equal to 1.24. Sorry, that should be -1.24. So, if we look at our values here, we see that When X is less than Minus 0.66, that the second derivative is negative, and when it's slightly greater than X equal to -0.66, it is the second derivative is positive. So we do have a sign change in the second derivative, so that means that X equal to minus the inverse hyperbolic tangent of the square root 3 divided by 3 is an inflection point. And if we look at the other inflection point, which is X is equal to 0.66, when we look at a value of X that is less than X equal to 0.66, we see that The second derivative is positive, and we look at a value of X that is slightly larger than X equal to 0.66, we see the second derivative is negative. So again, we do have a sign change at this point. So that means that the answer to this problem is that the points of inflection, the X values are X equal to plus or minus. The inverse hyperbolic tangent of the quantity of the square root of 3 divided by 3 in quantity. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.

Key Concepts

Points of Inflection

Second Derivative

Derivative of Hyperbolic Functions

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