Applications of Derivatives — Important Questions
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
45 important questions from Applications of Derivatives for Maharashtra Board Class 12 Mathematics & Statistics -Commerce, with answers.
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Important Questions from Applications of Derivatives
If the demand function is D = 100 - 2P, find the elasticity of demand when P = 20.
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2/3
Step 1: When P = 20, D = 100 - 2(20) = 60. Step 2: dD/dP = -2. Step 3: η = -P/D × dD/dP = -20/60 × (-2) = 40/60 = 2/3.
Which of the following statements about the function f(x) = x⁴ - 8x² + 5 are correct?
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f has minimum at x = ±2, f has maximum at x = 0, f is decreasing on (0, 2)
Step 1: f'(x) = 4x³ - 16x = 4x(x² - 4) = 4x(x-2)(x+2). Critical points: x = 0, ±2. Step 2: f''(x) = 12x² - 16. f''(0) = -16 < 0 (maximum), f''(±2) = 32 > 0 (minimum). Step 3: f'(x) < 0 on (0,2), so decreasing.
For what value of x does the function f(x) = x³ - 3x² + 2 have a local minimum?
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x = 2
Step 1: f'(x) = 3x² - 6x = 3x(x - 2). Critical points: x = 0, 2. Step 2: f''(x) = 6x - 6. Step 3: f''(0) = -6 < 0 (maximum), f''(2) = 6 > 0 (minimum). Local minimum at x = 2.
The cost function is C(x) = x² + 4x + 100. Find the marginal cost when x = 5.
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14
Step 1: Marginal cost = dC/dx = d/dx(x² + 4x + 100) = 2x + 4. Step 2: When x = 5, marginal cost = 2(5) + 4 = 14.
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