Share

Books Shortlist
Your shortlist is empty

# Solution for Find a Point on the Curve Y = X3 + 1 Where the Tangent is Parallel to the Chord Joining (1, 2) and (3, 28) ? - CBSE (Science) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?

#### Solution

​Let :  $f\left( x \right) = x^3 + 1$

The tangent to the curve is parallel to the chord joining the points  $\left( 1, 2 \right)$ and $\left( 3, 28 \right)$ .

Assume that the chord joins the points

$\left( a, f\left( a \right) \right)$ and $\left( b, f\left( b \right) \right)$ .
$\therefore$ $a = 1, b = 3$
The polynomial function is everywhere continuous and differentiable.
So,
$f\left( x \right) = x^3 + 1$ is continuous on $\left[ 1, 3 \right]$ and differentiable on $\left( 1, 3 \right)$ .
Thus, both the conditions of Lagrange's theorem are satisfied.
Consequently, there exists $c \in \left( 1, 3 \right)$ such that
$f'\left( c \right) = \frac{f\left( 3 \right) - f\left( 1 \right)}{3 - 1}$ .
Now,
$f\left( x \right) = x^3 + 1$
$\Rightarrow$ $f'\left( x \right) = 3 x^2$ ,
$f\left( 1 \right) = 2, f\left( 3 \right) = 28$
$\therefore$ $f'\left( x \right) = \frac{f\left( 3 \right) - f\left( 1 \right)}{3 - 1}$
$\Rightarrow$ $3 x^2 = \frac{26}{2} \Rightarrow 3 x^2 = 13 \Rightarrow x = \pm \sqrt{\frac{13}{3}}$
Thus,
$c = \sqrt{\frac{13}{3}}$  such that ​
$f'\left( c \right) = \frac{f\left( 3 \right) - f\left( 1 \right)}{3 - 1}$ .
Clearly,
$f\left( c \right) = \left[ \left( \frac{13}{3} \right)^\frac{3}{2} + 1 \right]$
Thus,
$\left( c, f\left( c \right) \right)$  i.e.​
$\left( \sqrt{\frac{13}{3}}, 1 + \left( \frac{13}{3} \right)^\frac{3}{2} \right)$  is a point on the given curve where the tangent is parallel to the chord joining the points $\left( 1, 2 \right)$ and $\left( 3, 28 \right)$ .
Is there an error in this question or solution?

#### APPEARS IN

Solution for question: Find a Point on the Curve Y = X3 + 1 Where the Tangent is Parallel to the Chord Joining (1, 2) and (3, 28) ? concept: Maximum and Minimum Values of a Function in a Closed Interval. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
S