Which point on the curve has the biggest slope?

Which point on the curve has the biggest slope? This is a calculus 1 optimization problem.

This problem is from Single Variable Calculus by James Stewart,

[nimmira]

I appreciate all of these videos, easy and hard. Been long time since I've graduated and it's good to get some knowledge back.
Not sure how some people think, but please keep in mind that he is doing math videos for all, on various levels. He is not doing these videos just to show off his skills with hard problems. I don't think these videos are for fun (well, they are at some point) but majorly educational. This type of problems which some call "boring" are really some critical basics in analysis, and one cannot read or spell words if one cannot sing or know the ABC.

[neilgerace]

A curve is steeper when the slope is -5 than when the slope is 2, but -5 < 2 ... Should the slope be the absolute value of the gradient?

[jonasaxelleporteza]

What if the question was asking for the tangent line with the steepest slope to an ellipse?

[lindiwemhlongo]

*It is 2025 now.. and your videos are still saving lives. Thank you sir for the great explanation.* 🙏😭

[Myquerencia]

Solve all the values of x and y such that 16^(x^2+y)+16^(y^2+x)=1

[aidorice]

Jacket so hard thank you for the help :)

[SlimThrull]

*scratches head* But points don't have a slope.

[adude]

I've never taken calculus but I know a bit about derivatives. I knew that I had to differentiate y and I did. But then all I did was write
- 15x^4 + 120x^2 = -15(x^4 - 8x^2)
And then complete the square
-15(x^4 - 8x^2 + 16 - 16) = -15((x^2 - 4)^2 - 16)
For maximum -t we need minimum t, which in this case was equivalent to minimum (x^2 - 4)^2, which is 0. Solving (x^2 - 4)^2 = 0 obviously gives x = +-2, and the slope was also 240. Here you saw a solution from a soon-to-be 10th grader

[atmonatmon]

Shoot a video about what is t: a ^ b = b ^ a * t

[JanekDarki]

I had this example on my exams, and I'm not quite sure if i did it well:
What are
monotonicity of functions:
f(x)= x+3/x+9/x^3+27/x^5...
I figure out, that this function if sum of: (3^n)*(x^(1-2n) but wasn't sure what to do next.

[guilhermefreire]

Well this is quite easy. I mean after the headache that were those crazy IIT JEE integrals.

[venuss.]

Thank you so much for the explanation. What is your native language? Curious.

[omopsingh]

From burger King to a mathematician .I salute Ur spirit (wish I knew ur name)

[SasquatchlifeX]

So I understand why we did the second derivative and found local max and min, but just by looking at the function, it would seem that the slope gets even larger at both negative and positive infinity. This graph is like the function x^3, it gets larger at both ends. Which would make the slope (tangent line) steeper.

Are those just local max values?

[AnonimityAssured]

It is interesting that "the largest slope" or "the biggest slope" does not mean the steepest slope, which might also be the most negative. By the way, the plural of "point of inflection/inflexion" is "points of inflection/inflexion".

[watermelon]

This is the biggest slope going up from the left to the right, but not going down.

[angelmendez-rivera]

Just to be clear, a positive slope is by definition bigger than a negative slope. Negative numbers are by definition smaller than positive numbers. I fail to see why people think a negative slope can be considered larger than a positive slope here.

[granhermon]

You should check the limits at infinity too, right?

[stevemonkey]

Does this not just solve for the greatest positive slope? Not the greatest absolute value slope?

[CubemasterXD]

Dont we have to check the x => infinity?