`f(x) = sinxsinhx - cosxcoshx , -4 <= x <= 4` Find any relative extrema of the function.

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This function is infinitely differentiable on entire `RR.` The necessary condition of extremum for such a function is `f'(x) = 0.`

To find the derivative of this function we need the product rule and the derivatives of sine, cosine, hyperbolic sine and hyperbolic cosine. We know them:)

So  `f'(x) = cosx sinhx + sinx coshx + sinx coshx - cosx sinhx = 2 sinx coshx.`

The function `coshx` is always positive, hence `f'(x) = 0` at those points where `sinx = 0.` They are `k pi` for integer `k,` and three of them are in the given interval: `-pi,` `0` and `pi.`

Moreover, `f'(x)` has the same sign as `sinx,` so it is positive from `-4` to `-pi,` negative from `-pi` to `0,` positive from `0` to `pi` and negative again from `pi` to `4.` Function `f(x)` increases and decreases accordingly, therefore it has local minima at `x=-4,` `x=0` and `x=4,` and local maxima at `x=-pi` and `x=pi.`

 

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