Using Rolle's theorem, find points on the curve y=16-x^2,x [-1,1]… - Vidyadip

Question

Using Rolle's theorem, find points on the curve $\text{y}=16-\text{x}^2,\text{x}\in[-1,1],$ where tagent is parallel to x-axis.

✓

Answer

The equation of the curve is,
$\text{y}=16-\text{x}^2\ ....(1)$
Let $P(x_1,y_1)$ be a point on it where the tangent is parallel to x-axis.
Then,
$\Big(\frac{\text{dy}}{\text{dx}}\Big)_\text{p}=0\ ....(2)$
Differentiating (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=-2\text{x}$
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)_\text{p}=-2\text{x}_1$
$\Rightarrow-2\text{x}_1=0$ (from(2))
$\Rightarrow\text{x}_1=0$
$P(x_1, y_1)$ lies on the curve $y = 16 - x^2$
$\therefore\text{y}_1=16-\text{x}_1^2$
When $x_1 = 0$,
$y_1 = 16$
Hence, $(0, 16)$ is the required point.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the principal value of the following:
$\sec^{-1}\Big(2\sin\frac{3\pi}{4}\Big)$

Write a value of $\int\text{e}^{\text{x}}\sec\text{x}(1+\tan\text{x})\text{dx}$

Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?

Evaluate the following definite integrals:
$\int_{0}^\limits{2\pi}\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{dx}$

A random variable X has the following probability distribution:

Values of X:	0	1	2	3	4	5	6	7	8
P(X)	a	3a	5a	7a	9a	11a	13a	15a	17a

Determine:
$\text{P}(\text{X}<3),\text{P}(\text{X}\geq3),\text{P}(0<\text{X}<5).$

Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is

Find the area of the parallelogram determinrd by the vectors:
$3\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}$ and $\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$

If the mean and variance of a binomial variate X are 2 and 1 respectively, find P (X > 1).

Solve the following equation:
$(\text{e}^\text{y}+1)\cos\text{x dx}+\text{e}^\text{y}\sin\text{x}\text{dy}=0$

Write the number of vectors of unit length perpendicular to both the vectors $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$and $\vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}}.$