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# Solution for Find a Point on the Curve Y = X2 + X, Where the Tangent is Parallel to the Chord Joining (0, 0) and (1, 2) ? - CBSE (Science) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?

#### Solution

​Let: $f\left( x \right) = x^2 + x$

The tangent to the curve is parallel to the chord joining the points $\left( 0, 0 \right)$ and $\left( 1, 2 \right)$ .

Assume that the chord joins the points

$\left( a, f\left( a \right) \right)$ and $\left( b, f\left( b \right) \right)$ .
$\therefore$ $a = 0, b = 1$
The polynomial function is everywhere continuous and differentiable.
So,
$f\left( x \right) = x^2 + x$ is continuous on $\left[ 0, 1 \right]$ and differentiable on $\left( 0, 1 \right)$ .
Thus, both the conditions of Lagrange's theorem are satisfied.
Consequently, there exists  $c \in \left( 0, 1 \right)$ such that
$f'\left( c \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0}$.
Now,
$f\left( x \right) = x^2 + x$
$\Rightarrow$ $f'\left( x \right) = 2x + 1$,
$f\left( 1 \right) = 2, f\left( 0 \right) = 0$
$\therefore$ $f'\left( x \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0}$
$\Rightarrow$  $2x + 1 = \frac{2 - 0}{1 - 0} \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}$
Thus,
$c = \frac{1}{2} \in \left( 0, 1 \right)$ such that ​$f'\left( c \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0}$ .
Clearly,
$f\left( c \right) = \left( \frac{1}{2} \right)^2 + \frac{1}{2} = \frac{3}{4}$.
Thus,
$\left( c, f\left( c \right) \right)$ i.e.​
$\left( \frac{1}{2}, \frac{3}{4} \right)$ is a point on the given curve where the tangent is parallel to the chord joining the points (4, 0) and (5, 1).
Is there an error in this question or solution?

#### APPEARS IN

Solution for question: Find a Point on the Curve Y = X2 + X, Where the Tangent is Parallel to the Chord Joining (0, 0) and (1, 2) ? concept: Maximum and Minimum Values of a Function in a Closed Interval. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
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