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Standard XII

Mathematics

Monotonically Increasing Functions

Let the funct...

Question

Let the function $y = f (x)$ be given by $x = t^{5} - 5 t^{3} - 20 t + 7$ and $y = 4 t^{3} - 3 t^{2} - 18 t + 3$ , where $t ϵ (- 2, 2)$ . Then $f^{^{'}} (x)$ at $t = 1$ is ?

\frac{5}{2}

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\frac{2}{5}

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\frac{7}{5}

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none of these

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Solution

The correct option is B $\frac{2}{5}$
Given, $x = t^{5} - 5 t^{3} - 20 t + 7$
$\frac{d x}{d t} = 5 t^{4} - 15 t^{2} - 20$
${(\frac{d x}{d t})}_{t = 1} = - 30$
Also, given $y = 4 t^{3} - 3 t^{2} - 18 t + 3$
$\frac{d y}{d t} = 12 t^{2} - 6 t - 18$
${(\frac{d y}{d t})}_{t = 1} = - 12$
So, ${(\frac{d y}{d x})}_{t = 1} = \frac{2}{5}$

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Similar questions

Q. If

y = f (x)

is represented as

x = g (t) = t^{5} - 5 t^{3} - 20 t + 7

and

y = h (t) = 4 t^{3} - 3 t^{2} - 18 t + 3, (- 2 < t < 2)

, then max

f (x)

is equal to

Q. A function y=f(x) is represented parainetrically as follows:

x = ϕ (t) = t^{5} - 5 t^{3} - 20 t + 7.

y = Ψ (t) = 4 t^{3} - 3 t^{2} - 18 t + 3.

Let the values of "t" at extrema of this function be t1 and t2.Find 2(t1+t2)? Note:

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Q. Let

f : [- \frac{1}{2}, 2] \to R

and

g : [- \frac{1}{2}, 2] \to R

be functions defined by

f (x) = [x^{2} - 3]

and

g (x) = | x | f (x) + | 4 x - 7 | f (x)

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Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by
$f (x) = [x^{2} - 3] and g (x) = | x | f (x) + | 4 x - 7 | f (x)$ ,where [y] denotes the greatest integer less than or equal to y for $y ϵ R$ . Then

Q. Let

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Let the function y=f(x) be given by x=t5−5t3−20t+7 and y=4t3−3t2−18t+3, where tϵ(−2,2). Then f′(x) at t=1 is ?

The correct option is B 25Given, x=t5−5t3−20t+7 dxdt=5t4−15t2−20(dxdt)t=1=−30Also, given y=4t3−3t2−18t+3dydt=12t2−6t−18(dydt)t=1=−12So, (dydx)t=1=25

Let the function $y = f (x)$ be given by $x = t^{5} - 5 t^{3} - 20 t + 7$ and $y = 4 t^{3} - 3 t^{2} - 18 t + 3$ , where $t ϵ (- 2, 2)$ . Then $f^{^{'}} (x)$ at $t = 1$ is ?