Q112 / 180 · COMEDK 2026 (9 May Shift 1) · Mathematics

If the function $f(x) = x^4 - 31x^2 + ax + 5$ has a turning point at $x = 1$, then the value of '$a$' is ___ and the function attains a ___ at $x = 1$

  • A
    $a = -50$ , local maxima
  • Correct answer
    B
    $a = 58$ , local maxima
  • C
    $a = 50$ , local minima
  • D
    $a = 58$ , local minima

Solution

Given $f(x) = x^4 - 31x^2 + ax + 5$

Differentiating with respect to $x$, we get:

$f'(x) = 4x^3 - 62x + a$

Since $f(x)$ has a turning point at $x = 1$, $f'(1) = 0$

$4(1)^3 - 62(1) + a = 0$

$a = 58$

Now, finding the second derivative:

$f''(x) = 12x^2 - 62$

At $x = 1$, $f''(1) = 12(1)^2 - 62 = -50$

Since $f''(1) < 0$, the function attains a local maxima at $x = 1$.

Answer: $a = 58$ , local maxima

Hence, the correct answer is option: 'B'.

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