Applied Mathematics

Calculus

Medium

applied

Let $f : R \to R$ be a function defined as $f(x) = 2x^3 - 21x^2 + 36x - 20$ , then :

(A) maximum value of $f(x)$ is $-3$

(B) minimum value of $f(x)$ is $-128$

(C) maximum value exists at $x = 6$

(D) minimum value exists at $x = 1$

Choose the correct answer from the options given below :

(A), (B) Only

(A), (B), (C) Only

(B), (C), (D) Only

(C), (D) Only

✅ Correct Option: 1

$f'(x)=6(x-1)(x-6)=0\Rightarrow x=1, 6$ .

$f''(1)=-30<0$ so max at $x=1$ : $f(1)=-3$ .

$f''(6)=30>0$ so min at $x=6$ : $f(6)=-128$ .

(A), (B) correct; (C), (D) wrong.

2023: 7 June Shift 1

The value of a for which the function $f(x) = a^x$ is increasing on R are given by :

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A product costs the manufacturer ₹ 20 per unit. The demand function is given by $p(x) = 1000 - 20x$ , then the quantity for maximum profit is :

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If $x = 2at, y = at^2$ , where 'a' is a constant, then $\frac{d^2 y}{dx^2}$ at $x = 2$ is :

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Match List - I with List - II.

List - I	List - II
(A) The minimum value of $f(x) = 8x^2 - 4x + 7$ is	(I) 48
(B) The maximum value of $f(x) = x + \frac{1}{x}$ , $x < 0$ is	(II) 13
(C) The maximum slope of the curve $y = -2x^3 + 6x^2 + 7x + 26$ is	(III) $-2$
(D) The minimum value of $f(x) = x^2 + \frac{128}{x}$ is	(IV) $\frac{13}{2}$

Choose the correct answer from the options given below :

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$\int_1^4 x^3\log x\, dx$ is equal to:

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The order and degree Of the differential equation $\frac{d^2 y}{dx^2} + 2 e^{-x} = 0$ , respectively are

Solution