Using Rolles theorem, find points on the curve `y=16-x^2` , `x in [-1,\ 1]` , where tangent is parallel to x-axis

Explanation

Solution:

The equation of the curve is `y=16-x^{2}.`

Let `{P}({x}_{1}, {y}_{1})` be a point on it where tangents is parallel to {x} - axis. Then, `(\frac{d y}{d x})_{p}=0 `.

Now, `{y}=16-{x}^{2} Rightarrow \frac{{dy}}{{dx}}=-2 {x} Rightarrow(\frac{{dy}}{{dx}})_{{p}}=-2 {x}_{1}`

Therefore, `(\frac{{dy}}{{dx}})_{{p}}=0 arrow-2 {x}_{1}=0={x}_{1}=0`


Since `P(x_{1}, y_{1})` lies on the curve `y=16-x^{2}`

Therefore,` y_{1}=16-x_{1}^{2}`

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