Question

Using Rolle's theorem, find points on the curve y = 16 − x², x ∈ [−1, 1], where tangent is parallel to x-axis.

Answer 1

The equation of the curve is

$$y=16-x^2$$  ...(1)

Let P$$(x_1,y_1)$$ be a point on it where the tangent is parallel to the x-axis .

Then,

$${\left(\frac{dy}{dx}\right)}_P=0$$   ...(2)

Differentiating (1) with respect to x, we get

$$\frac{dy}{dx}=-2x$$

$$\Rightarrow {\left(\frac{dy}{dx}\right)}_P=-2x_1$$

$$\Rightarrow -2x_1=0\left(\text{ from }\left(2\right)\right)$$

$$\Rightarrow x_1=0$$

$$P(x_1,y_1)$$  lies on the curve$$y=16-x^2$$ .

$$\therefore$$ $$y_1=16-{x_1}^2$$

When $$x_1=0$$ ,

$$y_1=16$$

Hence,$$(0,16)$$ is the required point .