JEE MainMathematicsModerateMonotonocity

If the function $f(x) = \frac{t + 3x - x^{2}}{x - 4}$ where
is a parameter has a local minima and a local maxima, then the
range of values of is.

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f(x) = \frac{t + 3x - x^{2}}{x - 4};f^{'}(x) = \frac{(x - 4)(3 - 2x) - \left( t + 3x - x^{2} \right)}{(x - 4)^{2}}

for maximum or minimum, f ' (x) = 0
- 2x^2 + 11x - 12 - t - 3x + x^2 = 0
- x^2 + 8x - (12 + t) = 0
for one M and m
D > 0
64 - 4(12 + t) > 0
16 - 12 - t > 0

\Rightarrow

4 > t or t < 4]

Source: Grass-G / M-12 Mathematics Paper 2 Nov.docx

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If the function f(x)=t+3x−x2x−4f(x) = \frac{t + 3x - x^{2}}{x - 4}f(x)=x−4t+3x−x2​ where is a parameter has a local minima and a local maxima, then therange of values of is.

Need a video for this idea?

If the function $f(x) = \frac{t + 3x - x^{2}}{x - 4}$ where
is a parameter has a local minima and a local maxima, then the
range of values of is.