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NVAMediumJEE 2025Applications of Derivatives (Monotonicity, Extrema)

JEE Mathematics 2025 Question with Solution

If the set of all values of aa, for which the equation 5x3−15x−a=05x^3 - 15x - a = 0 has three distinct real roots, is the interval (α,β)(\alpha, \beta), then β−2α\beta - 2\alpha is equal to _____

Answer

Correct answer:30

Step-by-step solution

Standard Method

Given: The equation is

5x3−15x−a=05x^3 - 15x - a = 0

Let

f(x)=5x3−15xf(x) = 5x^3 - 15x

Find: The value of β−2α\beta - 2\alpha where the set of all values of aa for which the equation has three distinct real roots is (α,β)(\alpha, \beta).

Differentiate f(x)f(x) to find the critical points:

f′(x)=15x2−15=15(x−1)(x+1)f'(x) = 15x^2 - 15 = 15(x - 1)(x + 1)

So, the critical points are

x=1 and x=−1x = 1 \text{ and } x = -1

Evaluate at turning points

Now evaluate the function at these points:

f(1)=5(1)3−15(1)=−10f(1) = 5(1)^3 - 15(1) = -10 f(−1)=5(−1)3−15(−1)=10f(-1) = 5(-1)^3 - 15(-1) = 10

For the cubic 5x3−15x−a=05x^3 - 15x - a = 0 to have three distinct real roots, the horizontal line y=ay = a must intersect the graph of y=f(x)y = f(x) at three distinct points. Hence,

a∈(−10,10)a \in (-10, 10)

Therefore,

α=−10,β=10\alpha = -10, \qquad \beta = 10

Now,

β−2α=10−2(−10)=10+20=30\beta - 2\alpha = 10 - 2(-10) = 10 + 20 = 30

Therefore, the value of β−2α\beta - 2\alpha is 3030.

Common mistakes

  • Using only f′(x)=0f'(x)=0 to find critical points but not evaluating f(x)f(x) at those points. This is wrong because the interval for aa depends on the local maximum and minimum values. Instead, compute f(1)f(1) and f(−1)f(-1) after finding the turning points.

  • Including the endpoints a=−10a=-10 and a=10a=10. This is wrong because at these values the cubic has a repeated root, so the roots are not distinct. Instead, use the open interval (−10,10)(-10, 10).

  • Confusing the equation 5x3−15x−a=05x^3-15x-a=0 with the graph y=5x3−15xy=5x^3-15x and forgetting that aa represents the horizontal line level. This leads to incorrect interpretation of the three-root condition. Instead, compare aa with the extreme values of f(x)f(x).

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