Name: Second Derivative Test
Uploaded: 2019-07-19
Duration: 12 min 48 s
Description: The maxima and minima of a function are the largest and smallest value of the function. Watch this amazing video to find the maxima and minima of a function.

Question

Find the maximum and minimum areas of the triangle formed by

x -

axis, tangent and normal at a point on the parabola

y = x^{2} + 1 (1 \leq x \leq 3)

Accepted Answer

Parabola $y = x^{2} + 1; 1 \leq x \leq 3$

Let point $P (t, t^{2} + 1)$

$\frac{d y}{d x} = 2 x \Rightarrow {(\frac{d y}{d x})}_{t, t^{2} + 1} = 2 t$

Equation of tangent

$y - (t^{2} + 1) = 2 t (x - t)$

It intersects at the point $B$ , put $y = 0$

Point $B = (t - \frac{t^{2} + 1}{2 t}, 0)$

Equation of normal $y - (t^{2} + 1) = - \frac{1}{2 t} (x - t)$

It intersects at a point $A$ , put $y = 0$

$x = t + 2 t (t^{2} + 1)$

$A = (t + 2 t (t^{2} + 1), 0)$

By distance formula

$A P = (t^{2} + 1) \sqrt{4 t^{2} + 1}$

Similarly

$B P = \frac{(t^{2} + 1)}{2 t} \sqrt{4 t^{2} + 1}$

Area $= \frac{1}{2} \times A P \times B P$

$A = \frac{1}{2} (t^{2} + 1) \sqrt{4 t^{2} + 1} \times \frac{(t^{2} + 1)}{2 t} \sqrt{4 t^{2} + 1}$

$A = \frac{{(t^{2} + 1)}^{2} (4 t^{2} + 1)}{4 t}$

$\frac{d A}{d t} = \frac{(2 (t^{2} + 1) (2 t) (4 t^{2} + 1) + {(t^{2} + 1)}^{2} (8 t)) 4 t - {(t^{2} + 1)}^{2} (4 t^{2} + 1) (4)}{16 t^{2}}$

$\frac{d A}{d t} = \frac{(t^{2} + 1) (20 t^{4} + 7 t^{2} - 1)}{4 t^{2}}$

Now $\frac{d A}{d t} > 0$ for $1 \leq t \leq 3$ $[∵ 1 \leq x \leq 3 \Rightarrow 1 \leq t \leq 3]$

So, area $A$ is increasing for $1 \leq t \leq 3$

So the maximum value of the area is for $t = 3$ and the minimum value of the area is for $t = 1$

The maximum area of $∆ A P B = \frac{925}{3}$ Sq. unit

The minimum area of $∆ A P B = 5$ Sq. unit

Find the maximum and minimum areas of the triangle formed by $x -$ axis, tangent and normal at a point on the parabola $y = x^{2} + 1 (1 \leq x \leq 3)$

Important Questions on Monotonicity, Maxima and Minima

Learn and Prepare for any exam you want !

Find the maximum and minimum areas of the triangle formed by x-axis, tangent and normal at a point on the parabola y=x2+11≤x≤3

Important Questions on Monotonicity, Maxima and Minima

Learn and Prepare for any exam you want !

Find the maximum and minimum areas of the triangle formed by $x -$ axis, tangent and normal at a point on the parabola $y = x^{2} + 1 (1 \leq x \leq 3)$