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The minimum value of the function y = 2x3 - 21x2 + 36x - 20 is:
- a)-120
- b)-126
- c)-128
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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The minimum value of the function y = 2x3 - 21x2 + 36x - 20 is:a)-120b...
Let's say that the function is y = f(x) = 2x3 - 21x2 + 36x - 20.
For maxima/minima points, f'(x) = 0.
⇒ 6x2 - 42x + 36 = 0
⇒ x2 - 7x + 6 = 0
⇒ x2 - 6x - x + 6 = 0
⇒ x(x - 6) - (x - 6) = 0
⇒ (x - 6)(x - 1) = 0
⇒ x - 6 = 0 or x - 1 = 0
⇒ x = 6 or x = 1.
Now, let's check these points for maxima/minima by inspecting the values of f''(x) at these points.
f''(6) = 12(6) - 42 = 72 - 42 = 30.
f''(1) = 12(1) - 42 = 12 - 42 = -30.
Since, f''(6) = 30 > 0, it is the point of minimum value.
And the minimum value is f(6):
= 2(6)3 - 21(6)2 + 36(6) - 20
= 432 - 756 + 216 - 20
= -128
For maxima/minima points, f'(x) = 0.
⇒ 6x2 - 42x + 36 = 0
⇒ x2 - 7x + 6 = 0
⇒ x2 - 6x - x + 6 = 0
⇒ x(x - 6) - (x - 6) = 0
⇒ (x - 6)(x - 1) = 0
⇒ x - 6 = 0 or x - 1 = 0
⇒ x = 6 or x = 1.
Now, let's check these points for maxima/minima by inspecting the values of f''(x) at these points.
f''(6) = 12(6) - 42 = 72 - 42 = 30.
f''(1) = 12(1) - 42 = 12 - 42 = -30.
Since, f''(6) = 30 > 0, it is the point of minimum value.
And the minimum value is f(6):
= 2(6)3 - 21(6)2 + 36(6) - 20
= 432 - 756 + 216 - 20
= -128
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The minimum value of the function y = 2x3 - 21x2 + 36x - 20 is:a)-120b)-126c)-128d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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