C3 increasing function

Hi, I differentiated a function, and my answer corresponds to the books. The gradient is $2sec^22x(sin2x + 1)$.

It then asks me to show it's an increasing function, and the book states f is defined for $\frac {-\pi}{4} <x< \frac {\pi}{4}$.

When x=$\frac {-\pi}{4}$, $2sec^22x$ is undefined as $\frac {2}{cos^22x} = \frac {2}{0}$ so as this is a multiple, isn't the whole gradient, at this value, undefined? Thus I'm not sure how to prove $2sec^22x(sin2x + 1) >0$

Hi, I differentiated a function, and my answer corresponds to the books. The gradient is $2sec^22x(sin2x + 1)$.

It then asks me to show it's an increasing function, and the book states f is defined for $\frac {-\pi}{4} <x< \frac {\pi}{4}$.

When x=$\frac {-\pi}{4}$, $2sec^22x$ is undefined as $\frac {2}{cos^22x} = \frac {2}{0}$ so as this is a multiple, isn't the whole gradient, at this value, undefined? Thus I'm not sure how to prove $2sec^22x(sin2x + 1) >0$

Yes the whole gradient at that value is undefined - but that's OK because the function itself is undefined there.

Hint: to show that ab > 0, show that a>0 and b>0.

Brianeverit & Smaug123 - thanks for the tips. Smaug123, when work out a>0 and b>0, do I put any values in?

You want to prove that the overall gradient is +ve.

You know that $sec^2 {2x}$ is always positive because it's a square - the only exception being when it's 0 and you've just demonstrated that this can't happen in the range given.

So the only thing you have to worry about is making sure that that second factor is +ve in the range given

Thanks Davros

I know within the range it will always be positive (within the brackets). But the reason I ask about the values is because I think I remember being shown a way to determine if a function is increasing or decreasing, without using values. Have I totally just fabricated this or is there a way?

Thanks Davros

I know within the range it will always be positive (within the brackets). But the reason I ask about the values is because I think I remember being shown a way to determine if a function is increasing or decreasing, without using values. Have I totally just fabricated this or is there a way?